Equivariant Sugrue conjecture on homological dimension of rational G-sheaves

Let $G$ be a profinite group acting on a $G$-space $X$. Suppose that $X$ has finite Cantor--Bendixson rank and non-empty perfect hull. Equivariant Sugrue conjecture. The homological dimension of rational $G$-sheaves over $X$ is infinite. This is an equivariant version of a conjecture of Sugrue concerning spaces with finite Cantor--Bendixson rank and non-empty perfect hull. The corresponding assertion for the space of closed subgroups, $X=\mathcal{S} G$, predicts that the category of rational Weyl-$G$-sheaves has infinite homological dimension.