Stapledon's effectiveness conjecture for equivariant Ehrhart series

Prove that for every lattice polytope with an affine action by a finite group, the equivariant h*-series is a polynomial if and only if it is effective.

Background

The paper defines the equivariant h*-series of a lattice polytope equipped with a finite-group action and calls it effective when all coefficients are actual group characters. Stapledon's effectiveness conjecture asserts an equivalence between polynomiality of the equivariant h*-series and effectiveness of its coefficients.

The paper explains that the implication from effectiveness to polynomiality is readily known, while the converse has been established only for selected classes, including simplices, hypercubes, permutahedra, hypersimplices under symmetric-group actions, and lattice polytopes admitting invariant lattice triangulations. The general conjecture therefore remains unresolved; the paper proves gamma-effectiveness for a particular class of order polytopes rather than resolving this conjecture in full generality.