HZ-graph classification equivalent to the Core Conjecture

Let $G$ be an HZ-graph, meaning a connected Class 2 graph with $\Delta(G_\Delta)\le2$, and let $P^*$ be the Petersen graph with one vertex removed. Let $\mathcal{O}_\Delta$ be the graph class defined in the paper. HZ-graph classification conjecture. If $G$ is an HZ-graph with maximum degree $\Delta$, then either $G\in\mathcal{O}_\Delta$, or $\Delta=2$ and $G$ is an odd cycle, or $\Delta=3$ and $G=P^*$. This is presented as an equivalent reformulation of the Core Conjecture and is therefore solved together with it.