Schroeder's genus conjecture for geodesic surfaces

Let $X$ be a geodesic surface of genus $g$, and let $c(X)$ denote its cop number. Schroeder's genus conjecture.

$$c(X)=O(\sqrt g).$$

The paper compares this conjecture with a linear upper bound and a square-root lower bound in the genus. It is proposed as a generalization of a conjecture for graphs embedded in surfaces and remains open.