Finite cop number for homogeneous game spaces

A metric space $(X,d)$ is homogeneous if for every $x,y\in X$ there is an isometry $i:X\to X$ with $i(x)=y$. Let $X$ be a game space, namely a compact geodesic metric space equipped with the Cops and Robber game. The homogeneity conjecture. If $X$ is homogeneous, then

$$c(X)<\infty.$$

This is posed as an open problem in the paper's discussion of links between cop number and metric structure; the source gives no resolution.