The lattice Cayley polytope conjecture

Let $\boldsymbol{\mathbf{P}}$ be a lattice $d$-polytope, and let $\operatorname{deg}_{\mathbbm{Z}}(\boldsymbol{\mathbf{P}})$ denote its lattice degree. A lattice Cayley polytope is a lattice polytope admitting a Cayley structure, with a specified length.

Lattice Cayley polytope conjecture. If

$$d>2\operatorname{deg}_{\mathbbm{Z}}(\boldsymbol{\mathbf{P}}),$$

then $\boldsymbol{\mathbf{P}}$ is a lattice Cayley polytope of length

$$d+1-2\operatorname{deg}_{\mathbbm{Z}}(\boldsymbol{\mathbf{P}}).$$

The claim is attributed in the source to Di Rocco, Haase, Nill, and Paffenholz and to Dickenstein and Nill; it is the lattice-polytope analogue motivating the combinatorial conjectures.