Göttsche–Kool universal-function conjecture for rank-two virtual Segre numbers

Let $S$ be the surface and $M$ the moduli space defined above, let $s\in\mathbb Z$, and let $\alpha\in K(S)$ have $\operatorname{rk}(\alpha)=s$. The universal functions $Q_s,R_s,T_s,V_s,W_s,X_s$ lie in $\mathbb C[[z]]$, while $S_s,S_{1,s},Y_s,Y_{1,s},Z_s,Z_{11,s}$ lie in $\mathbb C[[z^{\frac12}]]$. Set

$$z=t\left(1+\left(1-\tfrac{s}{2}\right)t\right)^{1-\frac{s}{2}}.$$

The functions $V_s,W_s,X_s,Q_s,R_s,T_s$ are given by the formulas in the conjecture, with $Y_s,Y_{1,s},Z_s,Z_{11,s},S_s,S_{1,s}$ algebraic functions. Göttsche–Kool conjecture. For every such $\alpha$, the virtual Segre series satisfies

$$\mathscr S_\alpha[p',x']=\mathscr U_\alpha[p',x']\Big|_{z^{\frac12\operatorname{vd} M}},$$

where

$$\begin{aligned} \mathscr U_\alpha[p',x']&=2^{2-\chi_{\rm h}+K^2}V_s^{c_2(\alpha)}W_s^{c_1(\alpha)^2}X_s^{\chi_{\rm h}}Y_s^{c_1(\alpha)K}Z_s^{K^2}e^{x'^2Q_s+c_1(\alpha)x'R_s+x'KS_s+p'T_s}\\ &\quad\times\sum_a(-1)^{ac_1}\mathcal{SW}(a)Y_{1,s}^{c_1(\alpha)a}e^{ax'S_{1,s}}Z_{11,s}^{a^2}. \end{aligned}$$

This is the rank-two adjustment of the Göttsche–Kool conjecture and gives a universal description of virtual Segre numbers in terms of surface invariants and Seiberg–Witten basic classes. The supplied text does not state whether the conjecture has been resolved.