The effectiveness conjecture for equivariant Ehrhart polynomials

Let $G$ be a finite group, let $M$ be a lattice of rank $d$, and let $\rho:G\to\operatorname{Aff}(M)$ be an affine representation. Let $P\subset M_\mathbb{R}$ be a $G$-invariant $d$-dimensional lattice polytope. The invariant $h^*(P,\rho;t)$ is the equivariant Ehrhart $h^*$-series associated with this data; it may be an effective virtual representation-valued series or a polynomial. The effectiveness conjecture. $h^*(P,\rho;t)$ is effective if and only if $h^*(P,\rho;t)$ is a polynomial. This conjecture remains open in general, although it has been verified in several cases, including when $d=2$, when $P$ is a simplex or hypercube, and in several specified group actions and classes of polytopes.