Schröder's cop-number conjecture for graphs of bounded genus

Let $G$ be a connected graph, let $g(G)$ be the smallest nonnegative integer such that $G$ can be drawn on an orientable surface of genus $g(G)$ without crossing edges, and define

$$c(g):=\max\{c(G)\mid g(G)=g\}.$$

Here $c(G)$ is the cop number of $G$.

Schröder's conjecture. For every $g\in\mathbb{N}$, we have

$$c(g)\leqslant g+3.$$

This would improve the best known general upper bound $c(g)\leqslant\left\lfloor 3g/2\right\rfloor+3$. The conjecture is known for $g\leqslant 3$, but remains open in general.