Jackson and Ringel's bichromatic number conjecture

Let $S_g$ be the orientable surface of genus $g$, and let $\beta(G)$ denote the smallest $h$ such that the graph $G$ is biembeddable in the orientable surface $S_h$. The bichromatic number of $S_g$, denoted by $\chi_2(S_g)$, is the maximum chromatic number over all graphs biembeddable in $S_g$. For $g\geq 1$, the Heawood-like inequality is

$$\chi_2(S_g) \leq \left\lfloor \frac{13+\sqrt{73+96g}}{2} \right\rfloor.$$

Jackson and Ringel's conjecture. For all $g\geq 1$, the bound in this inequality is tight:

$$\chi_2(S_g) = \left\lfloor \frac{13+\sqrt{73+96g}}{2} \right\rfloor.$$

This is a higher-genus generalization of the Map Color Theorem and the Earth-Moon problem. The source presents the claim as a conjecture; its resolution is not specified here.