Asymptotic structure. VI. Distant paths across a disc

Menger's theorem says that, for $k\ge0$, if $S, T$ are sets of vertices in a graph $G$, then either there are $k + 1$ vertex-disjoint paths between $S$ and $T$, or there is a set X of at most $k$ vertices such that every $S$-$T$ path passes through $X$. The ``coarse Menger conjecture'' proposed a generalization of Menger's theorem for paths that are far apart: for all $k, c$ there exists $\ell$, such that for every graph $G$ and subsets $S, T \subset V (G)$, either there are $k + 1$ paths between $S$ and $T$, pairwise with distance more than $c$, or there is a set $X \subset V (G)$ of at most $k$ vertices such that every $S$-$T$ path has distance at most $\ell$ from $X$. This is known to be false, but may be true if $G$ is planar. Here we show that it is true if $G$ is planar and all vertices in $S \cup T$ are on the infinite region. In this case, we also obtain a linear-time algorithm to test for the existence of $k+ 1$ paths between $S$ and $T$, pairwise with distance more than $c$.