Kainen's conjecture for complete graphs in arbitrary surfaces

For each nonnegative integer $g\neq 2$, let

$$M(g):=\left\lceil\frac{7+\sqrt{1+48g}}{2}\right\rceil$$

and let $S_g$ denote the orientable surface of genus $g$. A Kainen drawing is a drawing attaining Kainen's lower bound for the surface crossing number. Kainen's arbitrary-surface conjecture. For each nonnegative integer $g\neq 2$, there exists a Kainen drawing of the complete graph $K_{M(g)}$ in $S_g$. This proposed common generalization of the paper's theorem and the minimum-triangulation result remains a belief stated in the conclusion; no resolution is supplied in the provided text.