Universal formula conjecture for semistable Vafa–Witten invariants

Let $S$ be a smooth projective surface, let $P^\perp_{\alpha}(n)$ be the Joyce–Song pair invariant for charge $\alpha=(r,c_1(L),c_2)$, and let $\chi(\alpha(n))$ denote the relevant Euler characteristic. If $H^{0,1}(S)=0=H^{0,2}(S)$, then the universal formula conjecture. there exist rational numbers $\mathsf{VW}_{\alpha_i}(S)$ such that

$$P^\perp_{\alpha}(n)=\mathop{\sum_{\ell\ge 1,\,(\alpha_i=\delta_i\alpha)_{i=1}^\ell:}}_{\delta_i>0,\,\sum_{i=1}^\ell\delta_i=1} \frac{(-1)^\ell}{\ell!}\prod_{i=1}^\ell(-1)^{\chi(\alpha_i(n))}\chi(\alpha_i(n))\\;\mathsf{VW}_{\alpha_i}(S)$$

for $n\gg0$. When either $H^{0,1}(S)$ or $H^{0,2}(S)$ is nonzero, only the first term is taken:

$$P^\perp_{r,L,c_2}(n)=(-1)^{\chi(\alpha(n))-1}\chi(\alpha(n))\mathsf{VW}_{r,L,c_2}(S).$$

These identities relate pair invariants in the strictly semistable case to Vafa–Witten invariants and are proposed as universal formulae; the source provides no resolution status.