Three-matching intersection conjecture for perfect matching polytopes of small dimensions

In the study of Fulkerson’s conjecture, Fan and Raspaud (1994) proposed a weaker conjecture: in every bridgeless cubic graph G, there are three perfect matchings M1, M2, and M3 such that M1∩M2∩M3=0̸. In this note we prove that this conjecture is true if the dimension of the perfect matching polytope of G is at most 9. This includes the class of bridgeless cubic graphs with the property that all the vertices of the corresponding perfect matching polytope are affinely independent.