The genus-three formula conjecture for the coefficients $S_{n,k}^{(3)}$

Let $S_{n,k}^{(g)}$ denote the coefficients indexed by $n$, $k$, and genus $g$, and let the expression referred to as

$$be the proposed formula for these coefficients. **Genus-three formula conjecture.** The expression$$

, with $g=3$, gives $S_{n,k}^{(3)}$ for all $n$ and $k$. The formula agrees with all presently known experimental values and its sum over $k$ agrees with $B_n^{(3)}$; its validity beyond the tested range remains open.