Hilton–Zhao's vertex-splitting conjecture

Let $G^*$ be an $n$-vertex connected class 1 $\Delta$-regular graph with $\Delta>\frac{n}{3}$. A vertex-splitting replaces a vertex by two adjacent vertices whose neighborhoods partition the original neighborhood into two nonempty subsets; suppose $G$ is obtained from $G^*$ in this way. Vertex-splitting conjecture. Then $G$ is $\Delta$-critical. Hilton and Zhao proposed this conjecture in 1997. It has been verified in several denser cases, including $\Delta\geq \frac{3(n-1)}{4}$, while the full range $\Delta>\frac{n}{3}$ remains open.