Kirillov's unimodality conjecture for zig-zag poset chain polytopes

Let $\mathcal{P}_n\subset\mathbb{R}^n$ be the convex integral polytope defined by

$$x_i\geq 0\quad (1\leq i\leq n),\qquad x_i+x_{i+1}\leq 1\quad (1\leq i\leq n-1).$$

For its Ehrhart series, write

$$\sum_{m\geq 0}|m\mathcal{P}_n\cap\mathbb{Z}^n|t^m=\frac{\delta(\mathcal{P}_n;t)}{(1-t)^{n+1}},$$

which defines the Ehrhart $\delta$-polynomial $\delta(\mathcal{P}_n;t)$. A polynomial $f(t)=\sum_{i=0}^d a_i t^i$ is unimodal if there is an index $j$ such that $a_0\leq\cdots\leq a_j\geq\cdots\geq a_d$.

Kirillov's unimodality conjecture. For any $n\geq 1$, the Ehrhart $\delta$-polynomial $\delta(\mathcal{P}_n;t)$ is unimodal.

The conjecture concerns the coefficient shape of Ehrhart $\delta$-polynomials for the chain polytopes of zig-zag posets and arose in connection with Kostka and Catalan numbers. The supplied source describes the paper as proving this conjecture, so the conjecture is resolved.