Conjecture on super-maximized genera of graphs in surfaces

Let $V_c(g)$ be the maximum number of vertices of a graph of genus $g$ embedded in a surface, and let $b(g)$ be the stated upper bound for this quantity. For an integer $S$ satisfying $S\equiv 1$ or $3\pmod 6$, set

$$g=\frac{(S-1)(S-3)}{6}.$$

Super-maximization conjecture. One has

$$V_c(g)=b(g)$$

whenever $g=\frac{(S-1)(S-3)}{6}\geq 4$ and $S\equiv 1$ or $3\pmod 6$. This extends the theorem's verified infinite family of cases with $S\equiv 7\pmod {12}$, together with $S=9$, and asserts that all admissible genera in this parametrized family are super-maximized.