Stable-degree conjecture for permutation representations

Let $G$ be a finite group, let $\rho$ be a permutation representation of $G$, and let $P(\rho)$ be its permutation polytope. Set

$$d:={\rm \dim \,} P(\rho).$$

Two permutation representations are stably equivalent when they differ by the stabilization relation used in the paper.

Stable-degree conjecture. There exists a stably equivalent permutation representation $\rho'\colon G\to S_n$ such that

$$n\leq 2d.$$

The statement was checked for $d\leq 4$ in the paper and would imply the weak part of the strong embedding conjecture. Its general validity remains open.