The alternatingly increasing conjecture for IDP lattice polytopes

Let $P$ be a lattice polytope that is integrally closed (IDP), meaning that for every $m\in\mathbb{Z}_{>0}$ and every $x\in mP\cap\mathbb{Z}^n$ there exist $x^{(1)},\ldots,x^{(m)}\in P\cap\mathbb{Z}^n$ such that $x=x^{(1)}+\cdots+x^{(m)}$. Suppose that $P$ contains a lattice point in its relative interior. The Ehrhart $h^\ast$-polynomial $h^\ast(P;x)$ is defined by

$$\sum_{m\geq 0}\left|mP\cap\mathbb{Z}^n\right|x^m=\frac{h^\ast(P;x)}{(1-x)^{d+1}},$$

where $d$ is the dimension of $P$. Alternatingly increasing conjecture. The polynomial $h^\ast(P;x)$ is alternatingly increasing. The conjecture concerns the coefficient pattern of Ehrhart $h^\ast$-polynomials; the stated source attributes it to SV13, via BJM16. Its resolution is not specified in the supplied text.