Götsche–Kool–Laarakker horizontal universality conjecture for virtual Euler characteristics

Let $r>1$. The notation $\overline{\Delta}(q)$ and $\epsilon_r$ is as defined in the introduction. For a smooth polarized surface $(X,H)$ with $b_1(X)=0$ and $h^{2,0}(X)>0$, let $c_1\in H^2(X,\mathbb{Z})$ and $c_2\in H^4(X,\mathbb{Z})$, and assume that there are no rank $r$ strictly Gieseker $H$-semistable sheaves on $X$ with Chern classes $c_1,c_2$. Write $M_X^H(r,c_1,c_2)$ for the corresponding moduli space and $\mathrm{vd}(r,c_1,c_2)$ for its virtual dimension. Götsche–Kool–Laarakker conjecture. There exist universal power series

$$D_0,\\{D_{ij}\\}_{1\leq i\leq j\leq r-1}\in\mathbb{C}[[q^{1/(2r)}]]$$

such that $e^{\mathrm{vir}}(M_X^H(r,c_1,c_2))$ equals the coefficient of $q^{\mathrm{vd}(r,c_1,c_2)/(2r)}$ in

$$r^{2+K_X^2-\chi(\mathcal{O}_X)}\left(\frac{1}{\overline{\Delta}(q^{1/r})^{1/2}}\right)^{\chi(\mathcal{O}_X)}D_0^{K_X^2}\sum_{(a_1,\ldots,a_{r-1})\in H^2(X,\mathbb{Z})^{r-1}}\prod_i\epsilon_r^{ia_ic_1}\\,\mathrm{SW}(a_i)\prod_{i\leq j}D_{ij}^{a_ia_j}.$$

This is the horizontal universality conjecture for virtual Euler characteristics; the universal functions are known conjecturally in closed form for $r=2,3,5$, but the assertion in general remains open.