Menger's theorem for infinite graphs with ends

A well-known conjecture of Erd\H os states that, given an infinite graph
$G$ and sets $A,B\subseteq V(G)$, there exists a family of disjoint
$A$--$B$ paths $\mathcal P$ together with an $A$--$B$ separator $X$
consisting of a choice of one vertex from each path in $\mathcal P$.
There is a natural extension of this conjecture in which
$A$, $B$ and $X$ may contain ends as well as vertices.
We prove this extension by reducing it to the vertex version, which was recently
proved by Aharoni and Berger.

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