The module-theoretic reformulation of Oliver's conjecture

Let $p$ be an odd prime, let $G$ be a finite $p$-group, and let $V$ be a faithful $d53f_pG$-module. Call $V$ an $F$-module when it has an offender, namely an elementary abelian subgroup $1\neq E\leq G$ satisfying

$$\dim(V)-\dim(V^E)\leq \operatorname{rank}(E).$$

The module-theoretic reformulation. If $V$ is an $F$-module, then there exists $1\neq g\in\Omega_1(Z(G))$ such that the minimal polynomial of the action of $g$ on $V$ divides

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

The source describes this as a reformulation equivalent to Oliver's conjecture through the quotient $S/\mathfrak{X}(S)$; it is not stated as resolved.