The prime-power wreath-product surjective-strategy conjecture

Let $G$ be a group generated by elements of order $p^n$, let $H$ be a $p$-group acting faithfully on a set of switches $\Omega$, and let $G\wr_\Omega H$ denote the corresponding wreath product. Prime-power wreath-product conjecture. If $G$ can be generated by elements of order $p^n$, and $H$ is a $p$-group acting faithfully on the set of switches $\Omega$, then $G\wr_\Omega H$ has a surjective strategy. This conjecture combines the preceding $2$- and $3$-power examples into a broader proposed family; no general proof is given.