Tanaka–Thomas's Vafa–Witten invariant formula for Joyce–Song pairs

Let $Y$ be an Enriques surface with a generic polarization, let $v \in H^{\ast}(Y,\mathbb{Q})$, and let $P_v(n)$ be the equivariant-localization invariant of the moduli space of Joyce–Song pairs with Chern character $v$. Write $v(n)=v e^{c_1(\mathcal O_Y(n))}$ and $\chi(v)=\int_Y \operatorname{td}_Y v$.

Tanaka–Thomas's conjecture. There exist rational numbers $\mathsf{VW}(v_i)$ such that, for all $n\gg0$,

$$P_v(n)=\mathop{\sum_{\ell\ge1,\,(v_i=\delta_i v)_{i=1}^{\ell}:}}_{\delta_i>0,\,\sum_{i=1}^{\ell}\delta_i=1}\frac{(-1)^\ell}{\ell!}\prod_{i=1}^{\ell}(-1)^{\chi(v_i(n))}\chi(v_i(n))\,\mathsf{VW}(v_i).$$

This is a special case of the Tanaka–Thomas conjecture relating pair invariants to Vafa–Witten invariants. The passage from the pair invariant to the numbers $\mathsf{VW}(v_i)$ is asserted for sufficiently large $n$; no resolution is given in the source.