Gorenstein Fano Ehrhart root narrow-strip conjecture

Let $P$ be a Gorenstein Fano polytope of dimension $D$, and let $\alpha$ be a root of its Ehrhart polynomial. Gorenstein Fano Ehrhart root narrow-strip conjecture. All roots $\alpha$ of the Ehrhart polynomials of Gorenstein Fano polytopes of dimension $D$ satisfy

$$-\tfrac{D}{2}\leq\operatorname{Re}(\alpha)\leq\tfrac{D}{2}-1.$$

The conjecture narrows the general Ehrhart root strip for this important class of polytopes; computations for symmetric edge polytopes support it, while the general statement remains open.