\section*{Methods}

The samples we used were ultra-high mobility SiGe/Si/SiGe quantum wells. The peak electron mobility in these samples reaches 240~m$^2$/Vs. The approximately 15~nm wide silicon (001) quantum well was sandwiched between Si$_{0.8}$Ge$_{0.2}$ potential barriers. The samples were patterned in Hall-bar shapes with the distance between the potential probes of 150~$\mu$m and width of 50~$\mu$m using standard photo-lithography. The long side of the Hall bar corresponded to the direction of current parallel to the [110] or [$\bar1$10] crystallographic axis. The in-plane magnetic field was perpendicular to the current to exclude the influence of ridges on the quantum well surface on the resistance measured at high electron densities (for more details, see Refs.~\cite{melnikov2015ultra,melnikov2017unusual}). Measurements were carried out in an Oxford TLM-400 dilution refrigerator with a base temperature of $\approx 25$~mK. Data were taken by a standard four-terminal lock-in technique in a frequency range 0.5--11~Hz in the linear regime of response.

\subsection*{Data availability.} The data that support the findings of this study are available from the corresponding author upon reasonable request.





