A counterexample to the coarse Menger conjecture

Menger's well-known theorem from 1927 characterizes when it is possible to find k vertex-disjoint paths between two sets of vertices in a graph G. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the k paths are required to be pairwise at some distance at least d. The result is known for $k\le 2$, but we will show that it is false for all $k\ge 3$, even if G is constrained to have maximum degree at most three. We also give a simpler proof of the result when $k=2$.