The unimodality conjecture for $h^*$-vectors of IDP polytopes

Let $P$ be an IDP polytope, meaning a lattice polytope such that for every $k\in\mathbb{N}_{\ge 2}$, each lattice point in $kP$ is a sum of $k$ lattice points in $P$. If

$$h^*_P(t)=\sum_{i=0}^d h^*_i t^i$$

is the $h^*$-polynomial of $P$, then $(h^*_0,\ldots,h^*_d)$ is its $h^*$-vector. The unimodality conjecture. Every IDP polytope has a unimodal $h^*$-vector, meaning that there is some $j\in\{0,\ldots,d\}$ such that

$$h^*_0\le h^*_1\le\cdots\le h^*_j\ge h^*_{j+1}\ge\cdots\ge h^*_d.$$

This is described in the source as a main open question in Ehrhart theory. The paper reports counterexamples to log-concavity, but the stated unimodality conjecture remains open in the supplied context.