The Virasoro conjecture for stable sheaves on simply connected surfaces

Let $S$ be a smooth projective simply connected surface over $\mathbb C$, let $H$ be a fixed polarisation, let $r>0$, $c_2$ and a line bundle $L$ be chosen, and let

$$M=M_S^H(r,L,c_2)$$

be the moduli space of Gieseker semistable sheaves of rank $r$, determinant $L$, and second Chern class $c_2$. Assume that every semistable sheaf is stable and that $M$ has a (twisted) universal sheaf $\mathcal F$. Let $\mathbb D^S$ be the holomorphic descendents, let $\xi:\mathbb D^S\to H^\ast(M,\mathbb C)$ be their geometric realization, and let $\mathcal L_k$, for $k\geq -1$, be the Virasoro operators.

Virasoro conjecture. For every $k\geq -1$ and every $D\in\mathbb D^S$,

$$\int_{[M]^\mathrm{vir}}\xi_{\mathcal F\otimes(\operatorname{det}\mathcal F)^{-1/r}}\left(\mathcal L_kD\right)=0.$$

The conjecture is motivated by Virasoro constraints for Hilbert schemes and moduli spaces of sheaves. The source states that it is proved for K3 surfaces, while proposing it for simply connected surfaces in general.