Nguyen–Scott–Seymour's weak 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 $Y$. A set $Z\subseteq V(G)$ is $(a,b)$-centered if it is contained in the union of at most $a$ balls of radius at most $b$. Nguyen–Scott–Seymour's weak coarse Menger conjecture. There exist functions $f,g:\mathbb N\rightarrow\mathbb R$ 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 an $(f(k,r),g(k,r))$-centered set intersecting all $X$-$Y$ paths. The conjecture is a weakening of the strong coarse Menger conjecture, allowing both the number and radius of covering balls to depend on $k$ and $r$. Nguyen, Scott and Seymour later disproved it as well.