Ehrhart–Zeta duality under reversal of linear arbors

Establish that, for every linear arbor t, the Ehrhart polynomial E_t(u) of the lattice polytope Q_t equals the shifted Zeta polynomial Z_{Rev(t)}(u+1) of the poset P_{Rev(t)} associated with the reversed arbor Rev(t).

Background

A linear arbor is an arbor whose vertices form a single chain, and Rev(t) is obtained by reversing the order of its vertices. The paper develops recursive descriptions of both the Ehrhart polynomial of Q_t and the Zeta polynomial of P_t. The conjecture proposes a precise duality between these invariants after reversing the arbor and shifting the Zeta-polynomial variable. It would identify the normalized volume of Q_t with the number of maximal chains of P_{Rev(t)}. The conjecture was checked computationally for all linear arbors of size at most 11, but is not proved.