Etingof's generalized Benson conjecture for induced prime-to-\p symmetries

Let $p$ be a prime, let $G$ be a $p$-group, let $\Bbbk$ be an algebraically closed field of characteristic $p$, and let $V$ be an indecomposable representation of $G$. An induced $p'$-symmetry of $V$ is a pair $(\mathcal{C},G')$ as defined in the source, where $\mathcal{C}$ is the relevant tensor subcategory and $G'$ is the associated finite group.

Generalized Benson conjecture. If $(\mathcal{C},G')$ is an induced $p'$-symmetry of $V$, then every prime dividing the order of $G'$ is less than $p$.

The source presents this as an equivalent reformulation of Etingof's generalized Benson conjecture in the language of induced $p'$-symmetries. It gives computational evidence for the case $p=2$ but no resolution of the general statement.