Hilton's Overfull Conjecture for dense graphs

Let $G$ be a graph with vertex set $V(G)$, maximum degree $\Delta$, chromatic index $\chi'(G)$, and fractional chromatic index $\chi'_f(G)$. Hilton's Overfull Conjecture. If

$$\Delta>\frac{1}{3}|V(G)|,$$

then

$$\chi'(G)=\lceil \chi'_f(G)\rceil.$$

The conjecture predicts exact rounding of the fractional chromatic index for graphs whose maximum degree exceeds one third of their order. It remains wide open.