Stapledon's equivalence conjecture for equivariant H*-series

Let $G$ be a finite group acting on a lattice, and let $P$ be a $G$-invariant lattice polytope. Let $Y$ be the toric variety with ample line bundle $L$ associated to $P$, and let $H^*[t]$ denote the equivariant $H^*$-series of $P$. Stapledon's equivalence conjecture. The following conditions are equivalent: $L$ admits a $G$-invariant section defining a non-degenerate hypersurface of $Y$; $H^*[t]$ is effective; and $H^*[t]$ is a polynomial.

The implications from the first condition to the second and from the second to the third are known, but counterexamples show that neither the second nor the third condition implies the first. Thus the stated equivalence is refuted.