Gottsche–Kool monopole contribution conjecture for prime-rank refined Vafa–Witten invariants

Let $S$ and $c_1$ satisfy the hypotheses of Theorem 1 in the source, and let $r$ be prime. Write $\mathrm{VW}_{r,c_1,c_2}^{\mathrm{monopole}}(S,y)$ for the monopole contribution to the refined Vafa–Witten invariant and $\mathrm{VW}_{1^r,c_1,c_2}(S,y)$ for the contribution of the locus indexed by $1^r$. Gottsche–Kool's monopole contribution conjecture. For every $c_2\in\mathbb{Z}$, one has

$$\mathrm{VW}_{r,c_1,c_2}^{\mathrm{monopole}}(S,y)=\mathrm{VW}_{1^r,c_1,c_2}(S,y).$$

The conjecture is stated as the assertion that the $1^r$ contribution accounts for the entire monopole contribution. It has been proved by Thomas, so the claim is resolved for the stated hypotheses and prime ranks.