The subject of this work is the study of the Lovász-Schrijver PSD-operator LS+ applied to the edge relaxation ESTAB(G) of the stable set polytope STAB(G) of a graph G. We are interested in the problem of characterizing the graphs G for which STAB(G) is achieved in one iteration of the LS+-operator, called LS+-perfect graphs, and to find a polyhedral relaxation of STAB(G) that coincides with LS+(ESTAB(G)) and STAB(G) if and only if G is LS+-perfect. An according conjecture has been recently formulated (LS+-Perfect Graph Conjecture); here we verify it for the well-studied class of claw-free graphs.