Maximum ratio between limited-visibility capture and seeing numbers for visibility one

Determine the largest real number r for which there exists a graph G satisfying cop₁(G) ≥ r cop′₁(G), where cop₁(G) is the minimum number of visibility-one cops required to see the robber and cop′₁(G) is the minimum number required to capture the robber.

Background

The paper distinguishes between the visibility cop number cop_l(G), needed to guarantee seeing the robber, and the visibility capture number cop′_l(G), needed to guarantee capture. It proves that the difference cop_l(G)−cop′_l(G) can be arbitrarily large for every positive visibility parameter l.

For visibility l=1, the authors construct graphs with cop₁(G) ≥ 2cop′₁(G), and state that it is unclear whether this ratio can be made arbitrarily large. The subsequent problem asks for the largest attainable ratio, so the unresolved issue concerns multiplicative separation specifically in the visibility-one case.