Unimodality conjecture for matching polytopes of wheel graphs

Let $W_n$ be the wheel graph on $n$ vertices, with $n\geq 4$, and let $P_M(W_n)$ denote its matching polytope. The coefficients of the $h^*$-polynomial of $P_M(W_n)$ form its $h^*$-vector.

Unimodality conjecture. The $h^*$-vector of $P_M(W_n)$ is unimodal for every $n\geq 4$.

This conjecture is motivated by the unimodality of the computed $h^*$-vectors for $n=4,\dots,10$ and by the question of whether IDP polytopes have unimodal $h^*$-polynomials. The paper does not resolve it, and the broader question remains open.