The perverse Hodge generating-series conjecture for Enriques moduli spaces

Let $Y$ be a generic Enriques surface, let $\beta\in H_2(Y,\mathbb Z)$ be effective with $\beta^2=2d$ and $2\nmid\beta$, and let $M_d$ be one connected component of $M^Y(0,\beta,1)$. Write ${^ph}^{i,j}(M_d)$ for its perverse Hodge numbers and $H(M_d)$ for its Betti realization.

Enriques perverse Hodge generating-series conjecture. The perverse Hodge numbers satisfy the displayed generating-series identity involving the theta functions, eta functions, and the Betti realizations $H(M_d)$.

This is presented as a concrete form equivalent to the earlier perverse Hodge conjecture. It gives a fully explicit prediction for the refined curve-counting data of moduli spaces on generic Enriques surfaces; no resolution evidence is supplied.