The real-part bound for roots of Ehrhart polynomials

Let $P$ be a lattice $d$-polytope, and let $i_P(m)$ denote its Ehrhart polynomial. A complex number $\alpha$ is a root of $i_P$ when $i_P(\alpha)=0$.

Real-part bound. All roots $\alpha$ of Ehrhart polynomials of lattice $d$-polytopes satisfy

$$-d \leq \operatorname{Re} \alpha \leq d-1.$$

The conjecture is based on experimental data about complex roots and would sharpen the general norm and real-root bounds proved earlier in the paper. Its status is not specified in the supplied text.