The all-genus formula conjecture for the coefficients $S_{n,k}^{(g)}$

For each positive genus $g$, let $S_{n,k}^{(g)}$ be the coefficients indexed by positive integers $n$ and $k$, and let $\chi^{(g)}$ be a triangular array of constants. Let the expression referred to as

$$be the proposed formula for these coefficients. **All-genus formula conjecture.** The expression$$

, together with an appropriate triangular array of constants $\chi^{(g)}$, gives $S_{n,k}^{(g)}$ for all $n$, $k$, and $g>0$. This extends the formula beyond the verified cases $g=1,2$ and is compatible with the presently known experimental results for higher genus, but remains unproved.