Hahn's nonincreasing-distance conjecture for cop-win graphs

Let $G$ be an undirected cop-win graph, and consider optimal play, meaning that the cop minimizes capture time while the robber maximizes it. Hahn's conjecture. The distance between the cop and robber does not increase in consecutive rounds in optimal play.

This conjecture was posed by Hahn and published in B+13. It was refuted there.