MED decomposition conjecture for subcubic 2-connected graphs

Let $G$ be a 2-connected graph with maximum degree 3. A MED decomposition of $G$ is a decomposition of its edges into a matching, a union of disjoint even cycles, and an independent set of double-stars.

MED decomposition conjecture. Every 2-connected graph with maximum degree 3 has a MED decomposition.

MED decompositions are used to establish $(7,2)$-edge-choosability for several classes of cubic graphs, including many snarks. The source does not give a resolution of this conjecture.