Refined K3 multiple-cover conjecture for Vafa–Witten invariants

Let $S$ be a K3 surface, let $\alpha$ be a charge, and let $\chi_S(\alpha,\alpha)$ denote its Mukai pairing. Write $[r]_t=(t^{r/2}-t^{-r/2})/(t^{1/2}-t^{-1/2})$, and let $\operatorname{Hilb}^d S$ be the Hilbert scheme of $d$ points on $S$. Assume that $\mathcal O_S(1)$ is generic. Refined K3 multiple-cover conjecture.

$$\mathsf{VW}_{\alpha}(t)=\sum_{r\mid\alpha}\frac{t^{\chi_S(\alpha,\alpha)/(2r)-r}\chi_{-t^r}(\operatorname{Hilb}^{1-\chi_S(\alpha,\alpha)/(2r^2)}S)}{[r]_t^2}.$$

The formula refines the familiar numerical multiple-cover formula and was known in the prime multiple-cover case; the source proves that special case, while the general refined formula remains conjectural.