Integrality conjecture for Gopakumar–Vafa invariants of holomorphic symplectic 4-folds

Let $X$ be a holomorphic symplectic $4$-fold, let $\beta\beta\in H_2(X,\mathbb{Z})$ be a primitive curve class, and let $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$, $n_{1,\beta}(\gamma)$, and $n_{2,\beta}$ be the Gopakumar–Vafa invariants defined from the corresponding reduced Gromov–Witten invariants, for insertions

\gamma_i\in H^*(X,\mathbb{Z})$ and

gamma\in H^4(X,\mathbb{Z})$. Integrality conjecture. With these definitions,

$$n_{0,\beta}(\gamma_1,\ldots,\gamma_n),\,\,\, n_{1,\beta}(\gamma),\,\,\, n_{2,\beta}\in\mathbb{Z}.$$

The conjecture asserts that the rationally defined invariants have enumerative integrality, analogous to integrality conjectures for Gopakumar–Vafa-type invariants on Calabi–Yau varieties; its general status is not specified in the source.