The countable Erdős–Menger conjecture with ends

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, for countable graphs G, the extension of this conjecture in which A,B and X are allowed to contain ends as well as vertices, and where the closure of A avoids B and vice versa. (Without the closure condition the extended conjecture is false.)