The Sylow-versality conjecture for finite group actions

Let $G$ be a finite constant group, let $X$ be a $G$-variety, and let $G_p$ be a Sylow $p$-subgroup of $G$.

Sylow-versality conjecture. If $X$ is $G_p$-versal for every prime $p$, then $X$ is $G$-versal.

Every versal $G$-variety is $p$-versal for every prime, but the converse is not generally true; this conjecture asks whether versality for all Sylow subgroups is nevertheless sufficient for versality for the whole group.