Conjectural Zeta-polynomial generating series for Hochschild arbors

Derive or prove the conjectural generating-series identity for the refined Zeta polynomials of the arbor family t_n, namely 1 + ∑_{n≥1} 𝖹_{t_n}(u,1)s^n = exp(∫ u/((1−us)(1+s−us)) ds).

Background

For the arbor family t_n related to Hochschild polytopes, the authors use computer experimentation to guess generating functions for several invariants. The displayed identity for the refined Zeta polynomials is explicitly labeled conjectural and is not proved in the paper.

References

Using \verb|FriCAS| , one can guess the following conjectural formulas for various generating series in $s$ for the arbors $t_n$. For the Zeta polynomials of the posets $P_{t_n}$: $$1 + \sum_{n \geq 1} \mathsf{Z}_{t_n}(u, 1) sn \stackrel{?}{=} \exp\left( \int \frac{u}{(1-us)(1+s-us)} ds \right).$$

— On posets and polytopes attached to arbors  (2503.04247 - Chapoton, 6 Mar 2025) in Section 10, “Hochschild polytopes”