  Among the well-known sufficient degree conditions for the Hamiltonicity of a
finite graph, the condition of Asratian and Khachatrian is the weakest and thus
gives the strongest result. Diestel conjectured that it should extend to
locally finite infinite graphs~$G$, in that the same condition implies that the
Freudenthal compactification of $G$ contains a circle through all its vertices
and ends. We prove Diestel's conjecture for claw-free graphs.
