The degree conjecture for torus and higher-genus ramification polynomials

For $g=0$ and $m\ge 3$, or $g\ge 1$ and $m\ge 1$, let $f_m^{(g)}(\alpha_1,\ldots,\alpha_m)$ denote the rescaled ramification number associated with coverings of the sphere by a genus-$g$ surface. The degree conjecture for torus and higher-genus ramification polynomials. $f_m^{(g)}(\alpha_1,\ldots,\alpha_m)$ is a symmetric polynomial in $\alpha_1,\ldots,\alpha_m$ of total degree $m+3g-3$. This claim is supported by the explicit computations collected in the Appendix for the cases studied in the paper. The source gives no resolution beyond those computations.