Alternative weak cop number and excluded asymptotic minors

Determine whether every connected graph that excludes a finite planar graph as an asymptotic minor has finite alternative weak cop number $\wco'(G)$, and, if so, determine whether this number is bounded by a function of the maximum treewidth of finite asymptotic minors, possibly with the identity function as the bound.

Background

The alternative weak and strong games require the cops to prevent repeated visits to a finite subgraph rather than a bounded-radius ball. These variants coincide with the original games on locally finite graphs, but can differ on graphs with vertices of infinite degree. The question extends the asymptotic-minor problem to arbitrary connected graphs using $\wco'$.

References

Is it true that if a connected graph $G$ excludes some finite planar graph $H$ as an asymptotic minor, then $\wco'(G)< \infty$? If so, does there exist some function $f:\mathbb N\to \mathbb N$ such that $\wco'(G)\leq f\big(\max\sg{\tw(H): |H|<\infty~\text{and}~H\preceq_{\infty} G}\big)?$ Can we choose $f=\mathrm{id}_{\mathbb N}$?

— Coarse cops and robber in graphs and groups  (2502.15571 - Esperet et al., 21 Feb 2025) in Question (q: minors'), Section 5, subsection “Alternative versions of the weak and strong games”