The maximum matching conjecture for maximum edge-colorable subgraphs

Let $G$ be an $r$-regular graph with maximum degree $\Delta$ and edge-chromatic number $\chi'(G)=\Delta+1$. A maximum $\Delta$-colorable subgraph is a subgraph with as many edges as possible whose edge-chromatic number is at most $\Delta$, and let $M$ be a maximum matching of $G$. Maximum matching conjecture. There is a maximum $\Delta$-colorable subgraph $H$ of $G$ such that

$$M\cup E(H)=E(G).$$

This is proposed as a generalization of a theorem of Mkrtchyan and Steffen for bridgeless cubic graphs and perfect matchings. The conjecture asks whether the matching-completion property persists for regular class II graphs.