Aravind–Subramanian conjecture for the oriented chromatic number of bounded-genus graphs

Let $G$ be a graph of Euler genus $g$, and let $\chi_o(G)$ denote its oriented chromatic number.

Aravind–Subramanian conjecture. There is an absolute constant implicit in the $O$-notation such that

$$\chi_o(G)=2^{O\left(\sqrt{g}\right)}.$$

This removes the arbitrarily small positive term from the known bound $\chi_o(G)=2^{O(g^{1/2+\varepsilon})}$ for every constant $\varepsilon>0$.