The Erdős-Menger conjecture for source/sink sets with disjoint closures

Erdős conjectured that, given an infinite graph G and vertex sets A,B⊆V(G), there exist a set P of disjoint A-B paths in G and an A-B separator X 'on' P, in the sense that X consists of a choice of one vertex from each path in P. We prove the conjecture for vertex sets A and B that have disjoint closures in the usual topology on graphs with ends. The result can be extended by allowing A, B and X to contain ends as well as vertices.