  A simple graph $G$ with maximum degree $\Delta$ is overfull if $|E(G)|>\Delta
\lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_{\Delta}$, is the
subgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, the
chromatic index of $G$ equals $\Delta+1$ if $G$ is overfull. Conversely, Hilton
and Zhao in 1996 conjectured that if $G$ is a simple connected graph with
$\Delta\ge 3$ and $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ 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
$\Delta=3$ in 2003, and Cranston and Rabern proved the next case $\Delta=4$ in
2019. In this paper, we give a proof of this conjecture for all $\Delta\ge 4$.
