Goulden–Jackson polynomiality conjecture for single Hurwitz numbers

Let $g$ be fixed and let $\eta=(\eta_1,\ldots,\eta_n)$ be a ramification profile with $n=\ell(\eta)$. Set $r=2g-2+d+n$, where $d=\sum_i\eta_i$, and let $H_g(\eta)$ denote the corresponding single Hurwitz number. Goulden–Jackson polynomiality conjecture. For fixed $g$ and $n$, there is a symmetric polynomial $P_{g,n}$ in $\eta_1,\ldots,\eta_n$ such that

$$H_g(\eta)=r!\prod_i\frac{\eta_i^{\eta_i}}{\eta_i!}P_{g,n}(\eta_1,\ldots,\eta_n),$$

with $\deg P_{g,n}=3g-3+n$, no term of degree less than $2g-3+n$, and with the coefficient of every monomial of degree $d$ having sign $(-1)^{d-(3g+n-3)}$. This conjecture predicts a precise polynomial structure for higher-genus single Hurwitz numbers beyond the genus-zero formula. Its status is not specified in the source.