ELSV-type formula conjecture for one-part double Hurwitz numbers

For $g\geq 0$, $n\geq 1$, and $(g,n)\neq(0,1),(0,2)$, let $H^g_{(d),\beta}$ be the one-part double Hurwitz number, let $r^g_{(d),\beta}$ be its associated number of simple branch points, and suppose there is a moduli space $\overline{\operatorname{Pic}}_{g,n}$ with classes $\psi_i$ and $\Lambda_{2k}$. ELSV-type formula conjecture. For every such $g,n$,

$$H^{g}_{(d), \beta}=r^{g}_{(d), \beta}!\,d\int_{\overline{\operatorname{Pic}}_{g,n}}\frac{\Lambda_0-\Lambda_2+\cdots\pm\Lambda_{2g}}{(1-\beta_1\psi_1)\cdots(1-\beta_n\psi_n)},$$

where $\overline{\operatorname{Pic}}_{g,n}$ has a possibly virtual fundamental class of dimension $4g-3+n$, contains the stated Picard variety as an open subset, admits the stated forgetful morphisms and sections, its $\psi$-classes satisfy the stated pullback and intersection relations, and the $\Lambda$-classes are codimension-$2k$ Chern classes of a self-dual rank-$2g$ bundle with $\Lambda_0=1$. The formula is verified in the source for genus $0$ and genus $1$ with the specified choices of compactified Picard spaces, but remains conjectural in general.