The strong coarse Menger conjecture

Let $G$ be a finite or infinite graph, and let $X,Y\subseteq V(G)$. An $X$-$Y$ path is a path with at least one end in $X$ and at least one end in $Y$. For sets $S,T\subseteq V(G)$, let $\operatorname{dist}_G(S,T)$ be the infimum of the distances between their vertices. For $S\subseteq V(G)$ and $r\in\mathbb R$, let

$$N_G^{\leq r}[S]=\{v\in V(G):\operatorname{dist}_G(v,S)\leq r\}.$$

A set $Z\subseteq V(G)$ is $(k,r)$-centered if $Z\subseteq N_G^{\leq r}[W]$ for some $W\subseteq V(G)$ with $|W|\leq k$. The strong coarse Menger conjecture. There exists a real number $c>0$ such that for any graph $G$, subsets $X,Y$ of $V(G)$ and positive integers $k,r$, either $G$ contains $k$ $X$-$Y$ paths pairwise at distance at least $r$, or there exists a set $(k-1,cr)$-centered in $G$ intersecting all $X$-$Y$ paths. This is a large-scale analogue of Menger's theorem. The cases $k=2$ and arbitrary $r$ were proved, but Nguyen, Scott and Seymour disproved the case $k=r=3$, even for graphs of maximum degree at most $3$; consequently the conjecture is refuted.