Finite cop number for game spaces with finite doubling constant

A game space is a compact geodesic metric space equipped with the Cops and Robber game. Its doubling constant is the least $k\in\mathbb{N}\cup\{\infty\}$ such that every ball $B(x,r)$ can be covered by at most $k$ balls of radius $r/2$. The doubling conjecture. If $X$ is a game space with finite doubling constant, then

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

This is posed among the paper's open problems connecting cop number with classical metric properties; the source gives no resolution.