Toda's product formula conjecture for stable pair invariants

Let $X$ be the smooth projective variety under consideration, let $q$ and $t$ be formal variables, let $t^\beta$ record effective curve classes $\beta$, and write $\operatorname{PT}(X)$ for the stable-pair partition function. Let $n_g^\beta$ denote the integers appearing in the Gopakumar–Vafa formula. Toda's product formula conjecture. There exist integers $n_g^\beta\in\mathbb Z$, for $g\geq 0$ and $\beta\in H_2(X,\mathbb Z)$, such that

$$\operatorname{PT}(X)=\prod_{\beta>0}\left(\prod_{j=1}^{\infty}(1-(-q)^jt^\beta)^{j n_0^\beta}\cdot \prod_{g=1}^{\infty}\prod_{k=0}^{2g-2}(1-(-q)^{g-1-k}t^\beta)^{(-1)^{k+g}\cdot n_g^\beta\binom{2g-2}{k}}\right).$$

This is the product form of the stable-pair/Gopakumar–Vafa correspondence, relating stable-pair invariants to the BPS integers; it is presented in the source as Toda's conjecture.