A simple graph $G$ with maximum degree $Δ$ is overfull if $|E(G)|>Δ\lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_Δ$, is the subgraph of $G$ induced by its vertices of degree $Δ$. Clearly, the chromatic index of $G$ equals $Δ+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $Δ\ge 3$ and $Δ(G_Δ)\le 2$, then $χ'(G)=Δ+1$ implies that $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case $Δ=3$ in 2003, and Cranston and Rabern proved the next case $Δ=4$ in 2019. In this paper, we give a proof of this conjecture for all $Δ\ge 4$.