The lattice join projection conjecture

Let $\boldsymbol{\mathbf{P}}$ be a lattice $d$-polytope, and let $\operatorname{deg}_{\mathbbm{Z}}(\boldsymbol{\mathbf{P}})$ and $\operatorname{codeg}_{\mathbbm{Z}}(\boldsymbol{\mathbf{P}})$ denote its lattice degree and lattice codegree.

Lattice join projection conjecture. If

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

then there is a lattice projection mapping $\boldsymbol{\mathbf{P}}$ onto a lattice join of lattice polytopes having the same lattice codegree as $\boldsymbol{\mathbf{P}}$.

This is the lattice analogue of the stronger codegree-decomposition conjecture for point configurations and is motivated by the evidence for that conjecture.