Hibi–Tsuchiya's conjecture on the h-vector of cycle graph stable set Ehrhart rings

Let $G$ be a cycle graph with $\ell\geq 3$, let $R=E_{\mathbb{K}}[\operatorname{STAB}(G)]$ be its Ehrhart ring, and write its h-vector as $(h_0,h_1,\ldots,h_s)$ with $h_s\neq 0$. Since $\dim R=2\ell+2$ and $a(R)=-3$, one has $s=2\ell-1$. Hibi–Tsuchiya's conjecture. The h-vector of $R$ has the form

$$(1,h_1,h_2,\ldots,h_{\ell-1},h_{\ell-1}+(-1)^{\ell-1},h_{\ell-2}+(-1)^{\ell-2},\ldots,h_3-1,h_2+1,h_1,1).$$

The conjecture concerns the symmetry and alternating correction terms in the h-vector of the Ehrhart ring of a cycle graph. The source states that Hibi and Tsuchiya made this conjecture and that it is proved in the paper.