Enumerative interpretation of the KMK transform coefficients

Let $\Phi_g(c)$ denote the functions on the right-hand side of the main theorem, namely the coefficient functions in the expansion of the expected resolvent of the measures $\mu_\lambda$. The parameter $c=c(x^2)$ is as above.

Enumerative interpretation conjecture. The functions $\Phi_g(c)$ are generating functions for some family of branched coverings of the sphere, and the coefficients $\theta_g(k)$ enumerate a subfamily of them.

This conjecture proposes a geometric-combinatorial interpretation of the coefficients appearing in the fine structure of the KMK transform of the Poissonized Plancherel measure. The source gives no resolution of the conjecture.