The logarithmic-distance conjecture for localising a mouse on graphs

Let $G$ be a connected graph of order $n$. In the Cat and Mouse localisation game, the cat receives only the information specified by the game rules and seeks to determine a vertex within a prescribed distance of the mouse. The logarithmic-distance conjecture. The cat can localise the mouse up to distance $O(\log n)$ on any connected graph of order $n$. This conjecture was motivated by results for trees and grids, where the mouse can be localised within bounded distance under the stated game rules. The supplied text does not establish its resolution.