Jiang's multiple-cover conjecture for twisted Vafa–Witten invariants

Fix a $\mu_r$-gerbe ${\mathfrak S}\to S$, a class $\mathbf c$, and sufficiently large $m$. Let ${\mathcal P}^{\perp,\operatorname{tw}}_{\mathbf c}(m)$ be the twisted stable-pair invariant defined by virtual localization, and let $\mathbf c_i$ correspond to classes $\alpha_i=\delta_i\alpha$ with $\delta_i>0$ and $\sum_{i=1}^{\ell}\delta_i=1$.

Jiang's twisted multiple-cover conjecture. If $H^{0,1}(S)=H^{0,2}(S)=0$, there exist rational numbers $\operatorname{VW}^{\operatorname{tw}}_{\alpha_i}({\mathfrak S})$ such that

$${\mathcal P}^{\perp,\operatorname{tw}}_{\mathbf c}(m)=\sum_{\substack{\ell\geq1,\ (\alpha_i=\delta_i\alpha)_{i=1}^{\ell}:\\ \delta_i>0,\ \sum_{i=1}^{\ell}\delta_i=1}}\frac{(-1)^\ell}{\ell!}\prod_{i=1}^{\ell}(-1)^{\chi(\mathbf c_i(m))}\cdot\chi(\mathbf c_i(m))\cdot\operatorname{VW}^{\operatorname{tw}}_{\mathbf c_i}({\mathfrak S}).$$

When either $H^{0,1}(S)$ or $H^{0,2}(S)$ is nonzero, only the first term is taken:

$${\mathcal P}^{\perp,\operatorname{tw}}_{\mathbf c}(m)=(-1)^{\chi(\mathbf c(m))-1}\cdot\chi(\mathbf c(m))\cdot\operatorname{VW}^{\operatorname{tw}}_{\mathbf c}({\mathfrak S}).$$

This conjecture defines twisted Vafa–Witten invariants through a multiple-cover expansion of stable-pair invariants. The paper states that it proves equality of the relevant invariant theories for K3 gerbes, but the source does not explicitly provide a resolution status for the conjecture in this span.