Hibi–Ohsugi conjecture on unimodality of Gorenstein IDP polytopes

Let $P$ be a lattice polytope. It is Gorenstein if its associated Ehrhart ring is Gorenstein, and it has the integer decomposition property (IDP) if every lattice point in $mP$ is a sum of $m$ lattice points in $P$ for every positive integer $m$. Hibi–Ohsugi conjecture. If $P$ is Gorenstein and IDP, then its $h^*$-vector is unimodal. This would extend the known result that Gorenstein lattice polytopes with regular unimodular triangulations have unimodal $h^*$-vectors; the conjecture is presented as an open weakening of the unimodular-triangulation hypothesis.