Green–Héthelyi–Lilienthal reformulation of Oliver's p-group conjecture

Let $p$ be an odd prime and let $G$ be a finite $p$-group. Let $V$ be a faithful $\mathbb{F}_p[G]$-module. The module $V$ is an $F$-module for $G$ if there is a nontrivial elementary abelian subgroup $E\le G$ such that

$$|E|\,|C_V(E)|\ge |V|.$$

Let $\Omega_1(Z(G))$ denote the subgroup of central elements of order dividing $p$.

Green–Héthelyi–Lilienthal reformulation. If $V$ is an $F$-module, then there is an element $1\ne g\in\Omega_1(Z(G))$ such that the minimal polynomial of the action of $g$ on $V$ divides

$$(x-1)^{p-1}.$$

This is presented as a reformulation of Oliver's p-group conjecture. The source does not state a resolution, so its status is recorded as open.