Double Hurwitz topological-recursion conjecture

Let $d$ be a positive integer, let $q_1,\ldots,q_d$ be parameters, and set

$$P(z)=q_1z+q_2z^2+\cdots+q_dz^d.$$

Consider the rational spectral curve

$$x(z)=z\exp(-sP(z)),\qquad y(z)=P(z).$$

Let $DH_{g,n}(\mu_1,\ldots,\mu_n)$ denote the associated double Hurwitz numbers, and let $\omega_{g,n}$ be the correlation differentials produced by topological recursion. Double Hurwitz topological-recursion conjecture. For $(g,n)\ne(0,2)$, the expansions at $x_i=0$ satisfy

$$\omega_{g,n}=\sum_{\mu_1,\ldots,\mu_n=1}^{\infty}DH_{g,n}(\mu_1,\ldots,\mu_n)\prod_{i=1}^n\mu_i x_i^{\mu_i-1}\,\operatorname{d}x_i.$$

The conjecture proposes a topological-recursion description of double Hurwitz numbers, with evidence from the associated quantum curve and low-genus calculations. The source also says that it can be reduced to a weaker polynomial-structure conjecture; no resolution is supplied.