Minor-closedness conjecture for maximal cut-ideal generating degree

Let $G$ be a graph, let $I_G$ be its cut ideal, and let $\mu(I_G)$ denote the maximal degree of a binomial in a minimal generating set of $I_G$. A graph $H$ is a minor of $G$ if it is obtained from $G$ by deleting and contracting edges. Minor-closedness conjecture. For any $k$, the set of graphs $G$ such that

$$\mu(I_G) \leq k$$

is minor-closed. This would characterize graphs with bounded cut-ideal generating degree by finitely many excluded minors, in view of the Robertson–Seymour graph-minor theorem. The conjecture is supported by the fact that induced-subgraph formation and edge contraction correspond to faces of cut polytopes, and hence cannot increase generating degrees; deletion of edges is the remaining case, and its effect is not established here.