Matsui et al.'s root-bound conjecture for Gorenstein Fano polytopes

Let ${\mathcal P} \subset {\mathbb R}^d$ be a Fano polytope, meaning an integral convex polytope of dimension $d$ whose only interior integer point is the origin. Call ${\mathcal P}$ Gorenstein when its dual polytope is integral; equivalently, it is a reflexive polytope. Let $i({\mathcal P},n)$ be the Ehrhart polynomial of ${\mathcal P}$, and let $\alpha$ be one of its roots. Matsui et al.'s Gorenstein root-bound conjecture. All roots $\alpha$ of the Ehrhart polynomials of Gorenstein Fano polytopes of dimension $d$ satisfy

$$-\frac{d}{2} \leq \Re(\alpha) \leq \frac{d}{2}-1.$$

Because the Ehrhart polynomial of a Gorenstein Fano polytope satisfies $i({\mathcal P},n)=(-1)^d i({\mathcal P},-n-1)$, its roots are symmetric about $\Re(z)=-\frac12$; the conjecture predicts the sharper strip above. The paper verifies it when all roots are real or when $d\leq 5$, but the supplied status does not establish a complete resolution.