Dickenstein–Nill Cayley decomposition conjecture for lattice polytopes

Let $P$ be an $n$-dimensional lattice polytope, and let $cmu(P)$ denote its $cmathbb{Q}$-codegree. A Cayley sum is a polytope constructed by placing polytopes along the vertices of a standard simplex and taking their convex hull.

Dickenstein–Nill's Cayley decomposition conjecture. If

$$\mu(P) > \frac{n+1}{2},$$

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

$$\left\lfloor 2(n+1-\mu(P)) \right\rfloor.$$

The stronger conjecture by Dickenstein and Nill was disproved by Higashitani; the weaker version recorded here remains open. A slightly weaker statement, with the hypothesis $\mu(P) \geq \frac{n+2}{2}$, is known.