The torus ramification polynomial conjecture

Let $m\ge 1$, let $\alpha=(\alpha_1,\ldots,\alpha_m)$ be a ramification type, and let $n=\alpha_1+\cdots+\alpha_m$. For $i=1,\ldots,m$, let $e_i$ be the $i$-th elementary symmetric function in $\alpha_1,\ldots,\alpha_m$. The torus ramification polynomial conjecture.

$$f^{(1)}_m(\alpha)=\frac{1}{24}\left(n^m-n^{m-1}-\sum_{i=2}^m(i-2)!e_i n^{m-i}\right).$$

This conjecture gives a general explicit formula for the number of almost simple coverings of the sphere by a torus, extending the known formula for simple coverings. It is based on explicit computations for up to six ramification points; the source provides no resolution.