Batyrev–Nill codegree conjecture for Cayley decompositions

For a lattice polytope $P$, define its codegree by

$$\operatorname{codeg}(P)=\min_{\mathbb Z}\{t\mid tP\text{ has interior lattice points}\}.$$

For example, the unimodular simplex satisfies $\operatorname{codeg}(\Delta_n)=n+1$. Batyrev–Nill conjecture. There is a function $f(n)$ such that every $n$-dimensional polytope $P$ with $\operatorname{codeg}(P)\geqslant f(n)$ decomposes as a Cayley sum of lattice polytopes. This conjecture concerns the relationship between large codegree and Cayley-sum structure; it was later proven by Haase, Nill, and Payne, who showed in addition that $f(n)$ is at most quadratic in $n$.