The closed formula conjecture for genus-one ramification polynomials

For $m\ge 1$, let $e_i$ denote the $i$-th elementary symmetric function in the variables used to define $f_m^{(1)}$. The closed formula conjecture for genus-one ramification polynomials.

$$f^{(1)}_m=\frac{1}{24}\left(e_1^m-e_1^{m-1}-\sum_{i=2}^m(i-2)!e_i e_1^{m-i}\right).$$

This is a closed-form strengthening of the genus-one polynomial pattern inferred from the Appendix, analogous to the sphere formula $f_m^{(0)}=e_1^{m-3}$. The source provides no resolution.