Chapoton's simplicial-polytope conjecture for arbor polytopes

Let $\tau$ be an arbor on the ground set $[n]:=\{1,2,\dots,n\}$, and let $\mathcal Q_\tau$ be its associated $n$-dimensional arbor polytope. Define $h(\tau)=(h_i(\tau))_{0\leq i\leq n}$, where $h_i(\tau)$ is the number of points in $\mathcal Q_\tau\cap\mathbb N^n$ having exactly $i$ nonzero coordinates. Chapoton's conjecture. The vector $h(\tau)$ is equal to the $h$-vector of an $n$-dimensional simplicial polytope for every arbor $\tau$ of size $n$. In particular, $h(\tau)$ is palindromic and unimodal. This conjecture concerns the combinatorial structure of lattice-point enumerators of arbor polytopes; the supplied text records no resolution.