Matsui et al.'s root-bound conjecture for Ehrhart polynomials

Let ${\mathcal P} \subset {\mathbb R}^N$ be an integral convex polytope of dimension $d$, and let $i({\mathcal P},n)$ denote its Ehrhart polynomial. For a root $\alpha$ of $i({\mathcal P},n)$, write $\Re(\alpha)$ for the real part of $\alpha \in {\mathbb C}$. Matsui et al.'s root-bound conjecture. All roots $\alpha$ of the Ehrhart polynomials of integral convex polytopes of dimension $d$ satisfy

$$-d \leq \Re(\alpha) \leq d-1.$$

The conjecture proposes a uniform vertical strip containing all Ehrhart-polynomial roots; the paper's abstract discusses the analogous Gorenstein Fano restriction, while this unrestricted formulation is the claim stated here.