Stapledon's effectiveness conjecture for equivariant Ehrhart series

Let $P$ be a $G$-invariant lattice polytope, and let $Y$ be the corresponding toric variety with torus-invariant line bundle $L$. The equivariant $H^*$-series is the series $H^*(P;G)[t]$ with coefficients in the representation ring of $G$. Stapledon's effectiveness conjecture. The following conditions are equivalent: $(Y,L)$ admits a non-degenerate $G$-invariant hypersurface; the coefficients of $H^*(P;G)[t]$ are effective representations; and $H^*(P;G)[t]$ is a polynomial. The implication from polynomiality to effectiveness remains open in general and is known only in a handful of cases.