Dickenstein–Nill codegree conjecture for lattice polytopes

Let $P$ be an $n$-dimensional lattice polytope, and let $\operatorname{cd}(P)$ denote its codegree. A Cayley sum is a polytope obtained by assembling polytopes along the vertices of a simplex; a nontrivial Cayley structure has more than one summand.

Dickenstein–Nill conjecture. If

$$\operatorname{cd}(P)>\frac{n+2}{2},$$

then $P$ decomposes as a Cayley sum of lattice polytopes of dimension at most

$$2\bigl(n+1-\operatorname{cd}(P)\bigr).$$

This is the integral-toric counterpart of the Beltrametti–Sommese fibration conjecture. The source states that it remains open in its original generality, although it is proved for Gorenstein polytopes and under additional hypotheses.