Conjecture on the cop number of graphs embedded in non-orientable surfaces

Let $c(g)$ denote the maximum cop number of a graph embedded in an orientable surface of genus $g$, and let $\tilde c(g)$ denote the corresponding maximum for graphs embedded in a non-orientable surface of genus $g$. Surface cop-number conjecture.

$$\tilde c(g)=c(\lfloor g/2\rfloor).$$

The preceding argument establishes the upper bound $\tilde c(g)\leq c(g-1)$ using the orientable double cover, while the equality for all genera is presented as a conjecture and its status is not resolved in the supplied text.