\section{Supplemental Material}
\subsection{First principles theoretical approach}
We solve the linearized Boltzmann transport equation for phonons for the non-equilibrium distribution function, $n_\lambda = n^0_\lambda + n^1_\lambda$, resulting from a small applied temperature gradient, $\nabla T$, where $n^0_\lambda$ is the Bose distribution function, and $n^1_\lambda$ is the small deviation from $n^0_\lambda$ generated by $\nabla T$. Here, $\lambda\sim\left(j, \mathbf{q}\right)$ designates the phonon mode, where $j$ is the phonon branch and $\mathbf{q}$ is the phonon wave vector. The linearized Boltzmann equation is:
\begin{align}
\mathbf{v}_\lambda\cdot\nabla T\frac{\partial n^0_\lambda}{\partial T} = \frac{\partial n_\lambda}{\partial t}\Bigg|_{\mathrm{collisions}}	\label{SI:PBE}
\end{align}
where $\mathbf{v}_\lambda$ is the phonon velocity of the mode $\lambda$, and the right-hand side is the collision term, which includes three-phonon scattering, four-phonon scattering and phonon scattering by the isotopic mass disorder. Harmonic interatomic force constants (IFCs), and anharmonic third- and fourth-order IFCs are determined using the Density Functional Theory (DFT) as implemented in Quantum Espresso.  Expressing $n^1_\lambda = n^0_\lambda\left(n^0_\lambda + 1\right)\mathbf{F}_\lambda\cdot\left(-\nabla T\right)$ allows Eq.~\ref{SI:PBE} to be recast into an integral equation for the function, $\mathbf{F}_\lambda$, which is solved numerically. The thermal conductivity is then obtained as:
\begin{align}
\kappa_{\alpha\beta} = \frac{k_B}{V}\sum_\lambda\frac{\partial n^0_\lambda}{\partial T}v_{\lambda, \alpha}F_{\lambda, \alpha}
\end{align}
where $k_B$ is the Boltzmann constant, $V$ is the crystal volume, and $\kappa_{\alpha\beta}$ is the thermal conductivity tensor for heat current flow along the Cartesian direction, $\alpha$, resulting from a temperature gradient along the direction, $\beta$. For the cubic structures considered in the present work, the thermal conductivity tensor is diagonal: $\kappa_{\alpha\beta} = k\delta_{\alpha\beta}$. Further details of the computational approach including computation scheme for IFCs, expressions for the phonon scattering rates and implementation of the solution of the phonon Boltzmann equation are given in Ref.~\cite{ravichandran_unified_2018}. All the calculations in this study are performed using norm-conserving pseudopotentials with the local density approximation (LDA) for the exchange correlation. The converged parameters used in the first principles calculations in this work for BP and SiC, such as the energy cutoffs for the DFT calculations, $\mathbf{q}$-grids used in the solution of the Boltzmann equation and the cutoffs for the harmonic, cubic and quartic IFCs, can be found in Ref.~\cite{ravichandran_phonon-phonon_2020}. The pressure in our calculations is calculated by obtaining the derivative of the Helmholtz free energy, correct to fourth-order in anharmonicity ($F_{4^{\mathrm{th}}-\mathrm{order}}$), with respect to crystal volume ($V$) at each temperature ($T$) using the expression: $P(a) = -\frac{\partial F_{4^{\mathrm{th}}-\mathrm{order}}}{\partial V}\Bigg|_{T, a} \approx -\frac{F_{4^{\mathrm{th}}}(a + \Delta a) - F_{4^{\mathrm{th}}}(a - \Delta a)}{V(a + \Delta a) - V(a - \Delta a)}\Bigg|_T$, where $a$ is the crystal lattice constant and $\Delta a \sim 0.05\%$ of $a$. The complete expression for the Helmholtz free energy, correct to fourth-order in anharmonicity of the crystal potential, can be found in  section 10 of the supplemental materials in Ref.~\cite{ravichandran_non-monotonic_2019}.
\subsection{Supplemental figures to the main text for BP}
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.325]{SI_Fig1_Aug30_2020.png}
\end{center}
\caption{Process-wise three-phonon phase space of the acoustic phonons calculated for boron arsenide [BAs - \textbf{(a)}], cubic gallium arsenide [c-GaN - \textbf{(b)}], boron phosphide [BP - \textbf{(c)}] and silicon carbide [SiC - \textbf{(d)}], organized according to decreasing mass ratio of the heavy to light constituent atoms, which correlates with a decreasing size of the frequency gap between acoustic and optic phonons (A-O gap).  The four compounds, BAs, c-GaN, BP and SiC show similar sharp dips  in the phase space for scattering between three acoustic phonons (\emph{AAA} scattering, red stars).  The large A-O gap in BAs almost completely removes the \emph{AAO} scattering (black squares) and fully exposes the dip in the \emph{AAA} scattering phase space.  In contrast, \emph{AAO} scattering processes increasingly dominate in the frequency region of the \emph{AAA} dips in c-GaN, BP and SiC as the size of the A-O gap decreases.}	\label{SI:fig_1}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.4]{SI_Fig2_Aug29_2020.pdf}
\end{center}
\caption{\textbf{(a)} Schematic of phonon dispersions with a frequency gap between acoustic and optic phonons, $\Delta\omega_{A-O}$, and an optic phonon bandwidth of $\Delta\omega_O$. \textbf{(b)} Schematic of the corresponding three-phonon phase space for \emph{AAO} and \emph{AOO} processes illustrating the upper and lower cut-offs at $\Delta\omega_O$ and $\Delta\omega_{A-O}$, respectively.  If $\Delta\omega_{A-O} > \Delta\omega_O$, a frequency window opens in which only \emph{AAA} scattering can occur.}	\label{SI:fig_2}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.285]{SIFig_10BPvs11BP_PhonSpec.png}
\end{center}
\caption{\textbf{(a)} Phonon dispersions of $^{10}$BP and $^{11}$BP at ambient pressure. \textbf{(b)} Phase space for \emph{AAA} and \emph{AAO} three-phonon scattering processes for $^{10}$BP and $^{11}$BP. \textbf{(c)} Scattering rates for \emph{AAA} and \emph{AAO} three-phonon processes (top panel) and spectral contributions to $\kappa$ [$\kappa (\nu)$] (bottom panel) for $^{10}$BP and $^{11}$BP. The slightly stiffer optic phonons in $^{10}$BP compared to $^{11}$BP shifts \emph{AAO} processes to higher frequencies, thereby exposing more of the \emph{AAA} phase space dip, causing weaker \emph{AAO} scattering rates and, as a result, higher $\kappa$ contributions from a narrow frequency region (shown by the blue shaded region in \textbf{(c)}) in $^{10}$BP.}	\label{SI:fig_3}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.5]{SIFig_B11PvsMgO_100-300K_kOverk0_Aug30_2020Data.pdf}
\end{center}
\caption{Calculated $\kappa (P)$ of $^{11}$BP and MgO, scaled by their zero pressure values, $\kappa_0$, at temperatures of 100 K, 200 K and 300 K.  With decreasing temperatures, $\kappa (P)/\kappa_0$ decreases in $^{11}$BP, in striking contrast to the behavior in MgO which shows an increase in $\kappa (P)/\kappa_0$.}	\label{SI:fig_4}
\end{figure*}
\subsection{Increasing strength of four-phonon scattering in BP with increasing pressure}
Figs. 1 and 3(a) in the main text show that, as $P$ increases, inclusion of four-phonon scattering increasingly suppresses the $\kappa (P)$ of BP, and that, this suppression becomes even larger at higher $T$.  This behavior is driven by strengthening of \emph{AAAA} four-phonon scattering with increasing $P$, as shown in Fig.~\ref{SI:fig_5}.  Hydrostatic pressure stiffens the chemical bonding, resulting in increased interatomic force constants (IFCs).  These increases in the fourth-order IFCs produce corresponding increases in the four-phonon scattering matrix elements acting to increase the scattering rates of all four-phonon channels.  At the same time, the increases in second-order IFCs shift optic phonons to higher frequencies, thereby reducing their populations and reducing the scattering rates of all four-phonon processes involving optic phonons, such as \emph{AAAO}, \emph{AAOO} and \emph{AOOO} processes.  As a result, \emph{AAAA} four-phonon scattering rates are increased preferentially over all other four-phonon scattering rates, which contain at least one optic phonon. \\
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.4]{BP_4phProcwiseScatRate_300K_0to90GPa_4plots_InkscapeEdited.png}
\end{center}
\caption{Process-wise four-phonon scattering rates for BP at $T$ = 300 K at different pressures.  \emph{AAAA} scattering rates increase with increasing $P$ and dominate at high $P$. }		\label{SI:fig_5}
\end{figure*}
\pagebreak
\newpage
\clearpage
\subsection{Hidden influence of the selection rules on the $\kappa$ of SiC}
The results for the pressure and temperature-dependent phonon dispersions, three-phonon and four-phonon scattering rates, and $\kappa$ for SiC are presented in this section. Figure~\ref{SI:fig_6} (a) shows the stiffening of the longitudinal acoustic and all optic phonon branches, and the weak softening of the transverse acoustic phonon branch of SiC with pressure. Figure~\ref{SI:fig_6} (b) shows the evolution of the process-wise separated three-phonon and total four-phonon scattering rates with pressure at room temperature. The two plots show the striking similarity of the trends found for SiC and for BP (in the main text). Similarly, application of hydrostatic pressure causes a peak and drop in $\kappa$ in SiC (Fig.~\ref{SI:fig_7}) as in BP (in the main text). Furthermore, the observed temperature- and pressure-dependence of the competition between \emph{AAA} and \emph{AAO} scattering rates in BP is also seen in SiC (Fig.~\ref{SI:fig_8}). Finally, the dominant all-acoustic \emph{AAAA} four-phonon scattering channel (see Fig.~\ref{SI:fig_9}) drives the reduction in $\kappa$ of SiC at high $P$, also similar to BP.\\
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.255]{SIFig_SiC_DispersionPrDep_ScatRates_300K.png}
\end{center}
\caption{\textbf{(a)} Phonon dispersions for SiC at 300 K and different pressures. \textbf{(b)} Process-wise separated three-phonon and total four-phonon scattering rates of SiC at 300 K and different pressures.}	\label{SI:fig_6}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.275]{SIFig_SiC_kappa_34ph_IsoPureNat.pdf}
\end{center}
\caption{Pressure dependence of the calculated thermal conductivity of isotopically enriched and naturally occurring cubic SiC scaled by its zero pressure values for different temperatures.}	\label{SI:fig_7}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.5]{SIFig_SIC_3phProcwise_4phTotal_ScatRates_100K-1000K_0to90GPa.png}
\end{center}
\caption{Process-wise three-phonon and total four-phonon scattering rates for SiC at $T$ = 100 K and $T$ = 1000 K, and different pressures.}	\label{SI:fig_8}
\end{figure*}
\begin{figure*}[!ht]
\begin{center}
\includegraphics*[scale=0.4]{SiC_4phProcwiseScatRate_300K_0to90GPa_4plots_InkscapeEdited.png}
\end{center}
\caption{Process-wise four-phonon scattering rates for SiC at $T$ = 300 K and different pressures.  \emph{AAAA} scattering rates increase with increasing $P$ and dominate at high $P$, qualitatively similar to the behavior in BP.}	\label{SI:fig_9}
\end{figure*}
\end{document}