Reversal invariance of h-vectors for linear arbors

Prove that, for every linear arbor t, the h-vectors of the posets P_t and P_{Rev(t)} are equal.

Background

The poset P_t is graded by the coordinate-sum height function, and its h-vector is obtained from the associated cubical complex. Reversing a linear arbor changes the arbor structure and generally changes the corresponding polytope and poset, but the authors observe equality of the h-vectors in all tested cases. This conjecture was checked computationally for linear arbors of size at most 11 and is left unresolved.

References

For every linear arbor $t$, the $h$-vectors of $P_t$ and $P_{Rev(t)}$ are equal.

— On posets and polytopes attached to arbors  (2503.04247 - Chapoton, 6 Mar 2025) in Conjecture 4.2, Section 4, “Ehrhart-Zeta duality for linear arbors”