Hamiltonicity of 3-connected quasi-claw-free graphs

A graph G is called quasi-claw-free if it satisfies the property: d(x,y)=2⇒ there exists u∈N(x)∩N(y) such that N[u]⊆N[x]∪N[y]. Let G be a 3-connected quasi-claw-free graph of order n. If δ(G)⩾(n+5)/5, then G is hamiltonian.