Goulden–Jackson's structural conjecture for higher-genus Hurwitz generating series

Let

$$\phi_i(z,p)=\sum_{n\geq 1}\frac{n^{n+i}}{n!}p_nz^n,$$

where $i$ is an integer, and let $s$ be the unique formal power series in $x$ and $p$ satisfying

$$s=xe^{\phi_0(s,p)}.$$

For the Hurwitz generating series $H_g(x,p)$, and for a partition $\theta$ with size $|\theta|=n$ and length $l(\theta)$, Goulden–Jackson's conjecture. For $g\geq 2$,

$$H_g(x,p)=\sum_{e=2g-1}^{5g-5}\frac{1}{(1-\phi_1(s,p))^e}\sum_{n=e-1}^{e+g-1}\sum_{\substack{\theta\models n\l(\theta)=e-2(g-1)}}\frac{K^g_\theta}{\#\operatorname{Aut}(\theta)}\phi_{\theta_1}(s,p)\phi_{\theta_2}(s,p)\dots$$

for some rational numbers $K^g_\theta$. Here $H_g$ is the genus-$g$ part of the connected Hurwitz generating series, and the lower-genus cases $H_0$ and $H_1$ are already given separately. The conjecture predicts a precise rational form for all remaining genera, with coefficients depending only on $g$ and $\theta$.