The perverse Hodge number generating-series conjecture for Enriques moduli spaces

Let $Y$ be a generic Enriques surface, let $\beta\in H_2(Y,\mathbb{Z})$ be an effective curve class with $2\nmid\beta$, and let $M_{\beta}$ be one of the two connected components of the moduli space of one-dimensional stable sheaves on $Y$ with Euler characteristic $1$. Define

$${^ph}^{i,j}(M_{\beta}):= h^j\bigl({^p\mathcal H}^{i}(R\rho_*\mathbb{Q}[\dim M_{\beta}])\bigr),$$

where $\rho:M_{\beta}\to\mathbb{P}$ is the Hilbert–Chow morphism. If $\beta^2=2d$, write $b_{i,d}=b_i(M_{\beta})$.

The perverse Hodge number generating-series conjecture. The perverse Hodge numbers depend on $\beta$ only through $\beta^2$, and, writing ${^ph}^{i,j}(M_{\beta})={^ph}^{i,j}_d$, they satisfy the stated generating-series identity.

This conjecture refines the known fact that the Betti and Hodge numbers of $M_{\beta}$ depend only on the square of $\beta$. Its first assertion concerns deformation-invariant perverse data, while the explicit identity predicts all these numbers from the universal product and the ordinary Betti numbers.