Chapoton–Athanasiadis conjecture on h-polynomials of preorders

The conjecture concerns the identity $h(\tau)=h(\tau^*)$ for every preorder $\tau$, formulated in work of Christos A. Athanasiadis and Frédéric Chapoton. Their paper also records stronger conjectures about palindromicity, unimodality, $\gamma$-positivity, and real-rootedness.

Known results

• The identity and several related conjectures were verified computationally for preorders of size at most $7$.
• The conjectures other than real-rootedness were proved for chain preorders in earlier work cited as [Ath25].
• Further special cases were treated in [AXY26] and earlier work of Chapoton.

August 2026 claimed proof

Wang, Hou, Thawinrak, Liu, and Dai claim a general support-enumerator duality for bipartite graphs and identify it with the preorder $h$-polynomial identity. They derive lattice-point, Ehrhart, and volume formulas, but the supplied evidence contains no independent verification or peer review; the remaining conjectures are explicitly left open.

Current status (as of August 2026): The duality identity $h(\tau)=h(\tau^*)$ has a claimed general proof in a new preprint, but remains unverified, while the stronger conjectures remain open.