Conjecture on the top term of the generating function for double Hurwitz numbers

Let $s\geq 0$, let $\mu^{(1)},\ldots,\mu^{(s)}$ be partitions, let $l(\mu^{(i)})$ denote the length of $\mu^{(i)}$, and let $\#\operatorname{Aut}(\mu^{(i)})$ denote the order of its automorphism group. For the generating function $H_g(\mu^{(1)},\ldots,\mu^{(s)};T)$, write its top term as the leading singular term at $T=1$. Assume

$$5g+2l(\mu^{(1)})+\dots+2l(\mu^{(s)})\geq 6.$$

Top-term conjecture. The top term of $H_g(\mu^{(1)},\dots,\mu^{(s)};T)$ is

$$\frac{(5g+\sum_{i=1}^s 2l(\mu^{(i)})-7)!!\,c_g}{24^g(5g-3)!!\prod_{i=1}^s\#\operatorname{Aut}(\mu^{(i)})} \frac{\displaystyle\prod_{i=1}^s\prod_j\frac{(\mu^{(i)}_j)^{\mu^{(i)}_j}}{\mu^{(i)}_j!}}{(1-T)^{5g+\sum_{i=1}^s2l(\mu^{(i)})-5}}.$$

This conjecture is based on numerical experiments and predicts the leading singular behavior of the generating functions governing the relevant double Hurwitz numbers. The supplied text gives no resolution, so its status remains open.