Refined K3 Vafa–Witten generating-series conjecture

Let $S$ be a K3 surface, let $r$ be a positive integer, and let $\mathsf{VW}_{r,n}(t)$ denote the refined Vafa–Witten invariant for charge $(r,0,n)$. Let $[d]_t=(t^{d/2}-t^{-d/2})/(t^{1/2}-t^{-1/2})$, and define

$$\widetilde\Delta(q,t)=q\prod_{k=1}^{\infty}(1-q^k)^{20}(1-tq^k)^2(1-t^{-1}q^k)^2.$$

Assume that $\mathcal O_S(1)$ is generic. Refined K3 generating-series conjecture.

$$\sum_n\mathsf{VW}_{r,n}(t)q^n=\sum_{d\mid r}\frac{1}{[d]_t^2}\frac{d}{r}q^r\sum_{j=0}^{r/d-1}\widetilde\Delta\left(e^{2dj\pi i/r}q^{d^2/r},t^d\right)^{-1}.$$

This is the generating-series reformulation of the refined K3 multiple-cover conjecture; the general identity remains conjectural whenever it is not covered by the proved prime case.