The non-bipartite matching-polytope toric ideal degree conjecture

Let $G$ be a non-bipartite graph, let $\mathcal{M}_G$ be its matching polytope, and let $I_{\mathcal{M}_G}$ be the associated toric ideal. Write $\omega(I_{\mathcal{M}_G})$ for the maximal degree of a minimal generator of this ideal.

Matching-polytope degree conjecture. One has

$$\omega(I_{\mathcal{M}_G})\leq 4.$$

This conjecture extends the established bound $\omega(I_{\mathcal{M}_G})\leq 3$ for bipartite graphs; the paper gives non-bipartite examples attaining degree $4$, but the general upper bound remains open.