Götsche–Kool universal cobordism power-series conjecture

Let $S$ be a smooth projective surface with $b_1(S)=0$, $p_g(S)>0$, $K_S\ne0$, and only Seiberg–Witten basic classes $0$ and $K_S$. Let $H,c_1,c_2$ be chosen so that there are no rank $2$ strictly Gieseker $H$-semistable sheaves on $S$ with Chern classes $c_1,c_2$, put $M:=M_S^H(2,c_1,c_2)$, and let $\pi:M\to\mathrm{pt}$ be the projection. Götsche–Kool's universal cobordism conjecture. There exists a universal power series $L(p,\mathbf v)\in1+\mathbb Q[v_1,v_2,\ldots][[p]]$ such that $\pi_*[M]^{\mathrm{vir}}_{\Omega_*}$ is the coefficient of $p^{\operatorname{vd}(M)}$ in

$$\psi_S(p,\mathbf v)=8\left(\frac12\left(\sum_{n=0}^{\infty}[K3^{[n]}]p^{2n}\right)^{\frac12}\right)^{\chi(\mathcal O_S)}\left(2L(p,\mathbf v)\right)^{K_S^2}.$$

The conjecture extends the universality known for cobordism classes of Hilbert schemes of points and asserts that all dependence on $S$ occurs through $\chi(\mathcal O_S)$ and $K_S^2$; it remains open in the stated generality.