The Overfull Conjecture for graphs of maximum degree greater than one third their order

Let $G$ be a simple graph. Write $\Delta(G)$ for its maximum degree, and call $G$ overfull if

$$|E(G)|>\Delta(G)\left\lfloor\frac{|V(G)|}{2}\right\rfloor.$$

A graph is of Class 2 when its chromatic index satisfies $\chi'(G)=\Delta(G)+1$.

Overfull Conjecture. Let $G$ be a graph of Class $2$ with

$$\Delta(G)>\frac{|V(G)|}{3}.$$

Then $G$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$.

This conjecture of Chetwynd and Hilton would make determining the chromatic index polynomial-time solvable for graphs with maximum degree greater than one third their order. It is known in several dense cases, including when $\Delta(G)\geq |V(G)|-3$ and for sufficiently large graphs with $\Delta(G)\geq(1-\varepsilon)|V(G)|$ for $0<\varepsilon\leq 1/14$, but remains open in general.