Joyce–Song/GV special-case conjecture for stable pairs on Calabi–Yau fourfolds

In the setting of the wall-crossing conjecture, let $P^{\mathrm{JS}}_{n,\beta}(\gamma)$ denote the Joyce–Song chamber stable-pair invariant, and let $n_{0,\beta}(\gamma)$ be the genus $0$ Gopakumar–Vafa type invariant. Joyce–Song/GV special-case conjecture. For $n\geqslant1$,

$$P^{\mathrm{JS}}_{n,\beta}(\gamma)=\sum_{\substack{\beta_1+\cdots+\beta_n=\beta\\ \omega\cdot\beta_i=\frac{\omega\cdot\beta}{n},\ i=1,\ldots,n}}\prod_{i=1}^n n_{0,\beta_i}(\gamma),$$

while $P^{\mathrm{JS}}_{0,\beta}=P_{0,\beta}$. In particular, $P^{\mathrm{JS}}_{1,\beta}(\gamma)=n_{0,\beta}(\gamma)$.

This is a special case of the main wall-crossing conjecture in the Joyce–Song chamber. It is conjectural in the source and is checked in several examples.