The Ehrhart–Zeta duality conjecture for linear arbors

Let $t$ be a linear arbor, meaning an arbor in which each vertex has at most one sub-tree. Let $\operatorname{Rev}(t)$ be its reverse, obtained by putting the vertices in the opposite order. Let $Q_t$ be the associated polytope, $E_t(u)$ its Ehrhart polynomial, and $P_t$ its associated poset with Zeta polynomial $Z_t(u)$.

The Ehrhart–Zeta duality conjecture. For every linear arbor $t$,

$$E_t(u)=Z_{\operatorname{Rev}(t)}(u+1).$$

The conjecture would identify the normalized volume of $Q_t$ with the number of maximal chains of $P_{\operatorname{Rev}(t)}$. The source presents this as open and notes a further expected h-vector symmetry under reversal.