Stapledon's positive-coefficient conjecture for equivariant H*-polynomials

Let $P$ be a $G$-invariant polytope, and suppose that $H^*(P;G)[t]$ is an effective polynomial. Stapledon's positive-coefficient conjecture. If the coefficient of $t^m$ in the ordinary $h^*$-polynomial $h^*_P(t)$ is positive, then the coefficient of $t^m$ in $H^*(P;G)[t]$ contains the trivial representation with non-zero multiplicity. The conjecture is refuted: for $n\geq4$, the hypersimplex $\Delta_{2,n}$ supplies a counterexample, since its relevant ordinary coefficient is positive while the corresponding equivariant coefficient has zero trivial-representation multiplicity.