Versality detected by Sylow subgroups

Let $G$ be a finite group acting faithfully on a variety $X$. For each prime $p$, let $G_p$ be a Sylow $p$-subgroup of $G$. Sylow versality conjecture. The variety $X$ is $G$-versal if and only if, for every prime $p$, $X$ is $G_p$-versal. This would extend the corresponding result for complete non-singular toric varieties to arbitrary varieties, using that versality is a birational invariant.