The exceptional-graph conjecture for average color degree and rainbow matchings

Let $G$ be an edge-colored graph, let $\hat d(G)$ be its average color degree, and let $\widetilde{K_4}$ denote the exceptional edge-colored graph referred to in the source. A rainbow matching is a matching whose edges have pairwise distinct colors. Exceptional-graph conjecture. If $\hat d(G)\geq 2k-1$, then, unless $G$ is $\widetilde{K_4}$, the graph $G$ contains a rainbow matching of size at least $k$.

The authors state this as an affirmative expected answer to a question they were unable to resolve for all graphs. The precise definition of $\widetilde{K_4}$ is not included in the supplied context, so the exceptional case should be checked against the paper.