The proposed infinite-dimension criterion for equivariant dimensions

Let $\pi$ be a discrete group with an action by a group $\Gamma$, let $\pi\rtimes\Gamma$ be the associated semidirect product, and let $\mathcal{F}$ and $\mathcal{G}$ denote the families used in the paper, with $\mathcal{FIN}$ the family of finite subgroups. Infinite-dimension conjecture. If

$$\mathcal{F}\neq\mathcal{FIN},$$

then

$${\sf cd}_{\mathcal{G}}(\pi\rtimes\Gamma)={\sf gd}_{\mathcal{G}}(\pi\rtimes\Gamma)=\infty.$$

The claim concerns when the equivariant cohomological and geometric dimensions of the semidirect product become infinite. Because the supplied context does not define the families $\mathcal{F}$ and $\mathcal{G}$ or explain the status of this assertion, its interpretation and resolution require checking the source.