Stapledon's integrality conjecture for equivariant Ehrhart series

Let $G$ be a finite group acting on a lattice polytope $P\subseteq\mathbb{R}^n$. Let $M$ be the sublattice obtained by translating the affine span of $P$ to the origin, and let $\rho:G\to GL(M)$ be the induced representation. For $g\in G$, let $P^g$ be the fixed-point polytope and let $M^g$ be the sublattice of $M$ fixed by $g$. Define $\operatorname{ind}(P)$ as the smallest positive integer $k$ such that the affine span of $kP$ contains a lattice point, and define $\det(I-\rho(g))_{(M^g)^\perp}$ using the restriction of $\rho(g)$ to $(M^g)^\perp$. Stapledon's integrality conjecture. For every $g\in G$,

$$H^*[1](g)=\frac{\dim(P^g)!\cdot\operatorname{vol}(P^g)\cdot\det(I-\rho(g))_{(M^g)^\perp}}{\operatorname{ind}(P^g)}$$

is a non-negative integer. This conjecture predicts an arithmetic integrality property of the evaluation of the equivariant Ehrhart series and is part of Stapledon's program; the supplied text does not give a general resolution.