Hilton's conjecture on subgraph-overfull graphs

Let $G$ be a graph, let $V$ and $E$ denote its vertex and edge sets, and let $\Delta(G)$ be its maximum degree. A graph is Class 1 if its edges can be properly colored with $\Delta(G)$ colors; otherwise it is Class 2. A graph is subgraph-overfull if it has a subgraph $H$ satisfying $|V(H)|$ odd and

$$|E(H)|>\Delta(G)\frac{|V(H)|-1}{2}.$$

Hilton's conjecture. If $\Delta(G)>\frac{|V|}{3}$, then $G$ is Class 1 if and only if it is not subgraph-overfull. The supplied paper states that this conjecture holds for the class of split graphs with stretch index $\sigma=2$, but does not establish its general status.