Magic-positivity conjecture for preorder-polytope Ehrhart polynomials

Let $\tau$ be a preorder and let $\mathcal Q_\tau$ be its preorder polytope. Magic-positivity conjecture. The Ehrhart polynomial $\operatorname{Ehr}(\mathcal Q_\tau,t)$ is magic positive for every preorder $\tau$. In particular, $h^*(\mathcal Q_\tau,t)$ is real-rooted. The conjecture is known for arbor polytopes and for the standard Pitman--Stanley case, and has been verified computationally for all preorders of size at most $7$; the general case remains open.