Pandharipande–Thomas multiple-cover formula conjecture

Let $N_{n,\beta}$ denote the invariants appearing in the Gopakumar–Vafa-type product expansion of the stable-pair generating series, and let $(n,\beta)$ denote the common divisibility of $n$ and $\beta$. Pandharipande–Thomas multiple-cover conjecture. One has

$$N_{n,\beta}=\sum_{k\in\mathbb{Z}_{\ge1},\,k\mid(n,\beta)}\frac{1}{k^2}N_{1,\beta/k}.$$

The paper states that this is equivalent to the stronger rationality/product-form conjecture and that it implies more than the integrality conjecture; no resolution is supplied in the excerpt.