Mkrtchyan–Steffen decomposition conjecture for multigraphs

Let $G$ be a graph with maximum degree $4\Delta(G)=\Delta$4 and chromatic index $4\chi'(G)=\Delta+k$4$, where$k\geq 1$. A **maximum$4\Delta$4-edge-colorable subgraph** is a$4\Delta$4-edge-colorable subgraph containing as many edges as possible, and a decomposition partitions the edges of$G$into edge-disjoint subgraphs. **Mkrtchyan–Steffen's conjecture.**$G$can be decomposed into a maximum$4\Delta$4-edge-colorable subgraph$H_1$and a subgraph$H_2$such that$4\chi'(H_2)=k$4$. This generalizes their theorem for simple graphs to graphs with multiple edges; the source gives no resolution.