Quartic generation conjecture for cut ideals

Let $G$ be a graph and let $I_G$ be its cut ideal. A graph is $K_5$-minor-free when it has no minor isomorphic to $K_5$. Quartic generation conjecture.

$$I_G \text{ is generated in degrees }\leq 4\quad\Longleftrightarrow\quad G \text{ is }K_5\text{-minor-free}.$$

This extends the proposed characterization for quadratic generation. It is presented as a conjectural description of the complexity of Markov bases for graph cuts, and the authors also suggest that it captures the class of graphs with normal and Cohen–Macaulay cut ideals; that latter relationship is stated separately below.