Hilton–Zhao's Core Conjecture for class 2 graphs

Let $G$ be a simple connected graph with maximum degree $\Delta\ge 3$. The core $G_\Delta$ is the subgraph induced by the vertices of degree $\Delta$, and $\Delta(G_\Delta)\le 2$. A graph is overfull if $|E(G)|>\Delta\lfloor |V(G)|/2\rfloor$, and it is of class 2 when its chromatic index satisfies $\chi'(G)=\Delta+1$. Core Conjecture. If $G$ is of class 2, then $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting one vertex. Equivalently, every Hilton–Zhao graph $G\ne P^*$ with $\Delta(G)\ge 3$ is overfull. This conjecture would characterize class 2 connected graphs whose core has maximum degree at most $2$, apart from the exceptional graph $P^*$. The supplied text gives no evidence of a resolution, so its status is open.