Normality conjecture for cut polytopes of grid graphs

Let $G$ be a grid graph, and let ${\rm Cut}^\square(G)$ denote its cut polytope. Normality means that the affine semigroup generated by the homogenized cut vectors is saturated.

The grid-graph conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a grid graph.

The source presents this as another conjecture sufficient, together with the preceding reduction, for addressing the Sturmfels–Sullivant normality conjecture. No proof or resolution is given in the supplied text.