Song–Zhu's integral closedness conjecture for lattice polytopes

Let $M$ be a lattice and let $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$. For a lattice polytope $P\subseteq M_{\mathbb{R}}$, let $L(P)$ denote its lattice length, and write $rP\cap M$ for the lattice points of its $r$-fold dilation. The polytope is integrally closed if

$$rP\cap M=(P\cap M)+\cdots +(P\cap M)$$

for every $r\in\mathbb{Z}_{>0}$, with $r$ summands. Song–Zhu's conjecture. If $P\subseteq M_{\mathbb{R}}$ is an $n$-dimensional lattice polytope and $L(P)\ge n-1$, then $P$ is integrally closed. This is the toric analogue of projective-normality questions such as Mukai's conjecture. The paper presents it as a conjecture proposed by the first and third authors; the surrounding results provide evidence, but no resolution is given here.