Quasi-isometric bounded-treewidth characterization

Determine whether every locally finite graph with finite weak cop number is quasi-isometric to a graph of finite treewidth, and, if so, determine whether it is quasi-isometric to a graph of treewidth at most a function of its weak cop number, possibly with the bound f(k)=k+1.

Background

The paper proves that graphs quasi-isometric to graphs of bounded treewidth have bounded weak cop number, with an explicit upper bound in the locally finite setting. It also proves the exact characterization for weak cop number 1. The question asks whether these implications extend to all finite weak cop numbers.

References

Is it true that if a locally finite graph $G$ has finite weak cop number, then it is quasi-isometric to some graph of finite treewidth? If so, is it quasi-isometric to some graph of treewidth at most $f(\wco(G))$ for some $f:\mathbb N\to \mathbb N$? Can we choose $f(k)=k+1$?

— Coarse cops and robber in graphs and groups  (2502.15571 - Esperet et al., 21 Feb 2025) in Question (q: Qi-tw), Section 3.2, subsection “Graphs quasi-isometric to graphs of bounded treewidth”