Gao et al.'s conjecture on rainbow matchings in multigraphs

Let $G$ be an edge-coloured multigraph with $n$ colours such that each colour class is a matching of size $n+2$. A rainbow matching is a matching containing at most one edge of each colour. Gao et al.'s conjecture. The graph $G$ contains a rainbow matching of size $n$. This is a non-bipartite variant of the Aharoni–Berger problem, seeking a full rainbow matching when every colour class is sufficiently large; its general resolution is not supplied here.