Stapledon's equivariant volume conjecture

Let $P$ be a $G$-invariant lattice polytope. For each $g\in G$, define the evaluation $H^*(P;G)[1](g)$ of the equivariant volume at $g$. Stapledon's equivariant volume conjecture. For every $g\in G$, the value $H^*(P;G)[1](g)$ is a non-negative integer. If the $H^*$-series is effective and polynomial, then $H^*(P;G)[1]$ is a permutation representation. This conjecture concerns when equivariant volume is represented by an actual permutation action rather than merely a virtual representation; its general status is not established in the supplied context.