Sturmfels–Sullivant's normality conjecture for cut polytopes

Let $G$ be a finite graph without loops or multiple edges. For each subset $S$ of its vertex set, let $\delta_G(S)$ be the associated cut semimetric, let $A_G$ be the set of all such cut semimetrics, and let ${\rm Cut}^\square(G)=\operatorname{conv}(A_G)$ be the cut polytope. A lattice polytope is normal when its associated affine semigroup is saturated; equivalently here, if $X_G=\{(\mathbf a,1):\mathbf a\in A_G\}$, when

$$\mathbb Z_+(X_G)=\mathbb Z(X_G)\cap\mathbb Q_+(X_G).$$

Sturmfels–Sullivant's conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if and only if $G$ has no $K_5$ minor.

The conjecture is the central normality question for cut polytopes. The paper proves that normality is minor closed and establishes the conjecture for several classes of graphs, but states that the full conjecture remains open; it is enough to consider 4-connected plane triangulations.