Goulden–Jackson–Vakil conjecture for one-part double Hurwitz numbers

Let $h_{g;\mu_1,\ldots,\mu_n}^{\textnormal{one-part}}$ denote the double Hurwitz number with one ramification profile equal to the one-part partition of size $|\mu|=\mu_1+\cdots+\mu_n$. For integers $g\geq 0$ and $n\geq 1$ with $(g,n)\neq(0,1)$ or $(0,2)$, there should exist a moduli space $\overline{\operatorname{Pic}}_{g,n}$ with classes $\Lambda_{2k}\in H^{4k}(\overline{\operatorname{Pic}}_{g,n})$ and $\Psi_i\in H^2(\overline{\operatorname{Pic}}_{g,n})$ such that Goulden–Jackson–Vakil conjecture.

$$h_{g;\mu_1,\ldots,\mu_n}^{\textnormal{one-part}}=(\mu_1+\cdots+\mu_n)\int_{\overline{\operatorname{Pic}}_{g,n}}\frac{\sum_{k=0}^g(-1)^k\Lambda_{2k}}{\prod_{i=1}^n(1-\mu_i\Psi_i)}.$$

This conjecture seeks an ELSV-type intersection-theoretic formula for one-part double Hurwitz numbers, analogous to the formula for single Hurwitz numbers. The required compactified Picard-type moduli space and classes are part of the conjectural statement.