Structure morphism conjecture for the double Hurwitz moduli spaces

Let $\overline{P}(\mathbf{x})$ be the moduli space appearing in the proposed ELSV formula for double Hurwitz numbers, and let $\Lambda_{2g}$ be its tautological Chow class of degree $2g$. Let $\overline{\mathcal M}_{g,n}$ be the moduli space of stable $n$-pointed genus-$g$ curves, and let $\lambda_g$ denote its top Hodge class. Structure morphism conjecture. There is a natural structure morphism

$$\pi:\overline{P}(\mathbf{x})\to\overline{\mathcal M}_{g,n}$$

satisfying

$$\pi_*\Lambda_{2g}=\lambda_g.$$

The statement is offered as evidence for the proposed ELSV-type description, based on computations for one-part double Hurwitz numbers. Its status is not specified in the source.