Double Hurwitz coefficient-expansion conjecture

Let $DH_{g,n}(\mu_1,\ldots,\mu_n)$ be the double Hurwitz numbers, and let $d$ and $A_\mu^i$ be defined by

$$A_\mu^i=i\sum_{|\lambda|=\mu-i}\frac{\mu^{\ell(\lambda)-1}}{|\operatorname{Aut}\lambda|}q_{\lambda_1}q_{\lambda_2}\cdots q_{\lambda_{\ell(\lambda)}}s^{\ell(\lambda)},$$

where $\lambda$ is an integer partition with $\ell(\lambda)$ parts and $\operatorname{Aut}\lambda$ is the set of permutations of its parts that leave the tuple invariant. Double Hurwitz coefficient-expansion conjecture. For $2g-2+n>0$, there exist coefficients $C_{g,n}\big(\substack{i_1,\ldots,i_n\m_1,\ldots,m_n}\big)$ independent of $\mu_1,\ldots,\mu_n$ such that

$$DH_{g,n}(\mu_1,\ldots,\mu_n)=\sum_{i_1,\ldots,i_n=1}^d\prod_{k=1}^nA_{\mu_k}^{i_k}\sum_{m_1,\ldots,m_n=0}^{\mathrm{finite}}C_{g,n}\big(\substack{i_1,\ldots,i_n\m_1,\ldots,m_n}\big)\prod_{k=1}^n\mu_k^{m_k+1}.$$

This is presented as conjecturally equivalent to the polynomial-structure conjecture and as a route from linear loop equations to explicit double Hurwitz formulas; the source gives no resolution.