Dichotomy for cop numbers of finitely generated groups

Determine whether every finitely generated group has weak cop number and strong cop number each belonging to the set {1,∞}.

Background

The paper records a question raised by Lee, Martínez-Pedroza, and Rodríguez-Quinche, also cited by Cornect and Martínez-Pedroza, concerning whether finite cop numbers other than 1 can occur for finitely generated groups. The paper proves the weak-cop-number dichotomy for finitely presented groups, but does not resolve the question for all finitely generated groups or the corresponding strong-cop-number assertion.

References

Is it true that for any finitely generated group $\Gamma$ , we have $\wco(\Gamma)=1\text{ or }\infty$, and $\sco(\Gamma)=1\text{ or }\infty$?

— Coarse cops and robber in graphs and groups  (2502.15571 - Esperet et al., 21 Feb 2025) in Question 1.3, Section 1, subsection “Weak cop number 1”