Stanley's conjecture on unimodality of IDP polytopes

Let $P$ be a lattice polytope with the integer decomposition property (IDP), meaning that every lattice point in $mP$ is a sum of $m$ lattice points in $P$. Let $h^*_P(x)$ be its $h^*$-polynomial, and call its coefficient vector unimodal if it weakly increases and then weakly decreases. Stanley's conjecture. If $P$ is an IDP lattice polytope, then its $h^*$-vector is unimodal. This is a central unresolved question about coefficient inequalities for Ehrhart polynomials; the paper discusses counterexamples to several related variants, while the Gorenstein IDP case is known.