Sturmfels–Sullivant conjecture on normality and Cohen–Macaulayness of cut ideals

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. For each unordered partition $A|B$ of $V(G)$, let $q_{A|B}$ be a variable, and let $I_G$ be the kernel of the cut map defining the cut variety of $G$; thus $K[q]/I_G$ is its semigroup algebra. Sturmfels–Sullivant conjecture. The semigroup algebra $K[q]/I_G$ is normal if and only if $K[q]/I_G$ is Cohen–Macaulay if and only if $G$ is free of $K_5$ minors. This conjecture relates algebraic properties of cut ideals to the minor structure of the underlying graph; its resolution status is not determined by the supplied source information.