Equality of divergence and original cop numbers

Determine whether the divergence-game cop numbers equal the corresponding original cop numbers for every graph and, separately, for every group; specifically, determine whether $\sdi(G)=\sco(G)$ and $\wdi(G)=\wco(G)$, and whether $\sdi(\Gamma)=\sco(\Gamma)$ and $\wdi(\Gamma)=\wco(\Gamma)$.

Background

The divergence game changes the cops’ objective from eventually preventing repeated visits to a finite-radius ball to forcing the robber’s distance from a fixed vertex to tend to infinity. The paper proves equality at cop number 1, with strong equality characterized by hyperbolicity and weak equality by being quasi-isometric to a tree. It leaves equality for arbitrary cop numbers unresolved.