Normality conjecture for cut polytopes of 4-connected plane triangulations

Let $G$ be a 4-connected plane triangulation, 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 4-connected plane triangulation conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a 4-connected plane triangulation.

This is presented as a sufficient reduction for proving the Sturmfels–Sullivant conjecture, because the relevant class of graphs can be decomposed using graph sums and the remaining difficult cases are 4-connected plane triangulations. The source does not report a resolution.