The neatness conjecture for smooth lattice polytopes

Let $P\subset\mathbb{R}^n$ be a smooth lattice polytope. Write $P$ by integral inequalities $Ax\leqslant c$, and for $b\in\mathbb{Z}^m$ define its displacement by

$$P_b:=\{x\in\mathbb{R}^n:Ax\leqslant c+b\}.$$

The polytope $P$ is neat if whenever $P_b$ and $P_{-b}$ are normally isomorphic to $P$, one has

$$P_b\cap(-P_{-b})\cap\mathbb{Z}^n\ne\varnothing.$$

Neatness conjecture. All smooth lattice polytopes are neat.

The conjecture is motivated by the equivalence between neatness and preservation of weak or star Ewald conditions for lattice smooth bundles. The source explains that Oda's conjecture would imply neatness for all smooth lattice polytopes containing the origin, but the stronger assertion that every smooth lattice polytope is neat remains open.