The Robber Locating game

We consider a game in which a cop searches for a moving robber on a graph using distance probes, studied by Carragher, Choi, Delcourt, Erickson and West, which is a slight variation on one introduced by Seager. Carragher et al. show that for any fixed graph G there is a winning strategy for the cop on the graph G1/m obtained by replacing each edge of G by a path of length m, if m is sufficiently large. They conjecture that the cop does not have a winning strategy on Kn1/m if m