Ohsugi–Hibi conjecture on unimodality of Gorenstein IDP polytopes

Let $\mathcal{P}$ be a lattice polytope. It is Gorenstein if some positive integer dilate of $\mathcal{P}$ is reflexive, and it has the integer decomposition property if every lattice point in a positive integer dilate of $\mathcal{P}$ is a sum of lattice points from $\mathcal{P}$. The $h^\ast$-vector $h^\ast(\mathcal{P})$ is the coefficient vector in the numerator of the rational function representing the Ehrhart series of $\mathcal{P}$. Ohsugi–Hibi conjecture. If $\mathcal{P}$ is a Gorenstein polytope that has the integer decomposition property, then $h^\ast(\mathcal{P})$ is unimodal. This conjecture concerns the shape of Ehrhart $h^\ast$-vectors; it is attributed to Ohsugi and Hibi, and is a special case of conjectures of Brenti and Stanley on log-concavity and unimodality. It remains open according to the supplied source context.