The differential equation conjecture for weighted minimal d-Hurwitz generating functions

Let $d$ be a positive integer. For a partition $\alpha$ of $n$, let $h^d(\alpha)$ denote the minimal $d$-Hurwitz number, let $\mu^d(\alpha)$ denote its minimal factorization length, and write $p_\alpha$ for the product of the power-sum variables indexed by the parts of $\alpha$. Define

$$\widetilde{F}_d=\sum_{n \geq 1}\sum_{\alpha\vdash n}h^d(\alpha)\frac{z^n}{n!}\frac{u^{\mu^d(\alpha)}}{\mu^d(\alpha)!}p_\alpha.$$

Let $\widetilde{HW}([d])$ be the sum of the summations in the weighted $W$-operator $\widetilde{W}([d])$ whose degree is $d+1$. The differential equation conjecture. For every positive integer $d$,

$$\frac{\partial \widetilde{F}_d}{\partial u}=\widetilde{HW}([d])\widetilde{F}_d.$$

This conjecture proposes that differentiating the weighted generating function with respect to the factorization variable $u$ is governed precisely by the top-degree part of the weighted $W$-operator. The paper presents it as a generalization of the differential equation established for $d=3$; no resolution is given here.