The coarse Menger conjecture for graphs and geodesic metric spaces

Let $G$ be a graph or geodesic metric space, and let $X,Y\subseteq V(G)$ be two sets. For $k,d\in\mathbb{N}$, paths are pairwise $d$-separated when each two of them are at distance at least $d$.

Coarse Menger conjecture. There is a function $g:\mathbb{N}^2\to\mathbb{N}$ such that for every $k,d\in\mathbb{N}$, every graph or geodesic metric space $G$, and every two sets $X,Y\subseteq V(G)$, at least one of the following holds: there are $k$ disjoint $X$--$Y$ paths in $G$ that are pairwise at distance at least $d$ from each other; or there is a set $Z\subseteq V(G)$ of size at most $k-1$ such that every $X$--$Y$ path is at distance at most $g(k,d)$ from $Z$.

This is a coarse analogue of Menger's theorem, replacing disjointness and intersection by large separation and bounded distance. The conjecture is false in general, as indicated by the counterexamples discussed in the paper, but it holds for important classes including graphs whose cycle space is generated by cycles of bounded length, hyperbolic graphs, Cayley graphs of finitely presented groups, planar graphs with bounded face size, and complete Riemannian planes.