Stapledon's positivity conjecture for coefficients of equivariant Ehrhart series

Let $G$ be a finite group acting on a lattice polytope $P$, and let $H^*[z]=\sum_{j\geq 0}H^*_jz^j$ be its equivariant $H^*$-series. Let $h^*[z]$ be the ordinary $h^*$-polynomial of $P$. Stapledon's positivity conjecture. If $H^*[z]$ is a polynomial and the $i^{\text{th}}$ coefficient of $h^*[z]$ is positive, then the trivial representation occurs with non-zero multiplicity in the virtual character $H^*_i$. This conjecture predicts that positivity in the ordinary $h^*$-polynomial forces the corresponding equivariant coefficient to contain the trivial representation; the supplied text gives no resolution status for the general statement.