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We conjecture a formula for the virtual elliptic genera of moduli spaces ofrank 2 sheaves on minimal surfaces $S$ of general type. We express ourconjecture in terms of the Igusa cusp form $\chi_{10}$ and Borcherds type liftsof three quasi-Jacobi forms which are all related to the Weierstrass ellipticfunction. We also conjecture that the generating function of virtual cobordismclasses of these moduli spaces depends only on $\chi(\mathcal{O}_S)$ and$K_S^2$ via two universal functions, one of which is determined by thecobordism classes of Hilbert schemes of points on $K3$. We presentgeneralizations of these conjectures, e.g. to arbitrary surfaces with $p_g>0$and $b_1=0$.  We use a result of J. Shen to express the virtual cobordism class in terms ofdescendent Donaldson invariants. In a prequel we used T. Mochizuki's formula,universality, and toric calculations to compute such Donaldson invariants inthe setting of virtual $\chi_y$-genera. Similar techniques allow us to verifyour new conjectures in many cases.

A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula