The normality–$K_5$-minor conjecture for cut polytopes

Let $G$ be a graph, let ${\rm Cut}^\square(G)$ denote its cut polytope, and let $K[\operatorname{Cut}^\square(G)]$ denote the associated toric ring. A graph $H$ is a minor of $G$ if it can be obtained from $G$ by a sequence of edge deletions, edge contractions, and vertex deletions. The complete graph on five vertices is denoted by $K_5$.

Normality–$K_5$-minor conjecture. The following conditions are equivalent:

$${\rm Cut}^\square(G)\text{ is normal};$$ $$K[\operatorname{Cut}^\square(G)]\text{ is Cohen--Macaulay};$$ $$G\text{ has no }K_5\text{-minor}.$$

The conjecture extends the known implication that normality of ${\rm Cut}^\square(G)$ forces $G$ to have no $K_5$-minor. The converse, and hence the full equivalence, was stated as still open in the source.