Circumferences of 3-connected claw-free graphs

In Li et al. (2009), proved that a 3-connected claw-free graph of order nn with minimum degree δδ contains a cycle of length at least min{n,6δ−15}min{n,6δ−15}, and they conjectured that such graphs should have a cycle of length at least min{n,9δ−3}min{n,9δ−3}. We prove that this conjecture is true with δ≥8δ≥8.