The long-edge conjecture for the integer decomposition property

A lattice polytope is the convex hull of finitely many points in ${\mathbb Z}^d$, and it has the integer decomposition property if, for every positive integer $m$ and every lattice point $x\in mP\cap{\mathbb Z}^d$, there exist lattice points $x_1,\ldots,x_m\in P\cap{\mathbb Z}^d$ such that

$$x=x_1+\cdots+x_m.$$

The edge length of an edge is its length with respect to the lattice.

Long-edge conjecture. Simple lattice polytopes with sufficiently long edges have the integer decomposition property, where the required meaning of “sufficiently long” is given by an invariant uniform in the dimension.

This conjecture proposes that the integer decomposition property follows from independent lower bounds on the edge lengths, generalizing results for lattice polytopes whose edge lengths have a common factor. It was proved by Gubeladze in a precise form.