Alternative weak cop number and quasi-isometric bounded treewidth

Determine whether every graph with finite alternative weak cop number $\wco'(G)$ 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 $\wco'(G)$, possibly with the bound f(k)=k+1.

Background

The paper establishes an upper bound $\wco'(G)\leq \tw(G)+1$ for connected graphs without a local-finiteness assumption. The question asks whether finite alternative weak cop number characterizes, up to quasi-isometry, graphs of finite treewidth and whether a quantitative bound is possible.

References

Is it true that if a graph $G$ is such that $\wco'(G)<\infty$, 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 5, subsection “Alternative versions of the weak and strong games”