Mohar's cop-number conjecture for geodesic surfaces

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

$$c(S)=O(\sqrt{g}).$$

The paper describes this as a tough conjecture because it implies the corresponding conjecture for graphs of genus $g$. It is presented as open, with the known upper bound only linear in $g$ and lower bounds of order at least $g^{\frac{1}{2}-o(1)}$.