Broersma's induced-net Hamiltonicity conjecture

Let $G$ be a graph, and call the connected graph with degree sequence $3,3,3,1,1,1$ a net, with its vertices of degree $1$ called endvertices. Call $G$ claw-free if it has no induced $K_{1,3}$, and let $G$ have order $n$. Broersma's conjecture. If $G$ is $2$-connected and every endvertex of each induced net in $G$ has degree at least $\frac{n-2}{3}$, then $G$ is hamiltonian. The conjecture is a degree condition for Hamiltonicity in claw-free graphs and generalizes classical Hamiltonicity results. It was proved in the paper containing this statement.