Götsche–Kool conjecture for arbitrary surfaces with holomorphic 2-forms

Let $a\in\{\operatorname{cob},\operatorname{ell}\}$, let $S$ be a smooth projective surface with $b_1(S)=0$ and $p_g(S)>0$, and choose $H,c_1,c_2$ so that there are no rank $2$ strictly Gieseker $H$-semistable sheaves on $S$ with first Chern class $c_1$. Put $M:=M_S^H(2,c_1,c_2)$. Götsche–Kool's conjecture for arbitrary surfaces with holomorphic $2$-forms. The coefficient of $p^{\operatorname{vd}(M)}$ in $\mathsf Z^a_{S,c_1}(p,u^a)$ equals the coefficient of $p^{\operatorname{vd}(M)}$ in

$$\psi^a_{S,c_1}(p,u^a)=4\left(\frac12F^a_0(p,u^a)\right)^{\chi(\mathcal O_S)}\left(2F^a_1(p,u^a)\right)^{K_S^2}\sum_{b\in H^2(S,\mathbb Z)}\operatorname{SW}(b)(-1)^{c_1b}\left(\frac{F^a_1(-p,u^a)}{F^a_1(p,u^a)}\right)^{bK_S}.$$

Here $F^a_0,F^a_1$, $\mathsf Z^a_{S,c_1}$, and $\operatorname{SW}(b)$ are the series and Seiberg–Witten invariants used in the paper. The conjecture generalizes the rank $2$ formula from surfaces with only two basic classes to arbitrary surfaces with a holomorphic $2$-form, and is related to a formula of Dijkgraaf–Park–Schroers; it remains open in general.