Every 3-connected, essentially 11-connected line graph is Hamiltonian

Thomassen conjectured that every 4-connected line graph is Hamiltonian. A vertex cut X of G is essential if G−X has at least two non-trivial components. We prove that every 3-connected, essentially 11-connected line graph is Hamiltonian. Using Ryjáček's line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is Hamiltonian.