The coarse connectivity witness conjecture

Let $K,n,m,r\in\mathbb{N}$. A graph is $(M,A)$-quasi-isometric to a graph of tree-width at most $g$ in the usual coarse sense, and connected sets are at least $K$ apart when every pair of vertices from distinct sets has distance at least $K$. Then there exist some $M,A,g\in\mathbb{N}$ such that every graph with no $(M,A)$-quasi-isometry to a graph of tree-width at most $g$ contains connected sets $U_1,\ldots,U_n$ that are pairwise at least $K$ apart and such that for every pair $1\leqslant i<j\leqslant n$, there exist no $m$ balls of radius at most $r$ hitting all paths between $U_i$ and $U_j$. Coarse connectivity witness conjecture.

This proposes a coarse analogue of the connectivity-witness step in proofs of the classical grid theorem. The paper explains that the coarse Menger step fails, but leaves this alternative structural statement open.