The genus-square-root conjecture for proper chromatic number

Let $G$ be a graph embedded on an orientable surface of genus $g$. A proper orientation is an orientation in which adjacent vertices have different outdegrees, and the proper chromatic number $\Vec{\chi}(G)$ is the minimum possible maximum outdegree over all proper orientations of $G$.

Genus-square-root conjecture. If $G$ is a graph of genus $g$, then

$$\Vec{\chi}(G)\le O(g^{1/2}).$$

The paper previously obtains the weaker bound $O(g^{1/2}\log g/\log\log g)$ and proposes this conjecture as a possible improvement; the sharper square-root bound remains open.