LMOV integrality conjecture for open topological strings

Let $X$ be a toric Calabi–Yau threefold with brane configuration $\mathcal{D}$, let $\mathcal{P}_+$ denote the relevant set of nonempty partitions, and write the open-string free energy as

$$F_{str}^{(X,\mathcal{D})}(g_s,a,\mathbf{x})=\sum_{\mu\in \mathcal{P}_+}F_{\mu}^{(X,\mathcal{D})}p_{\mu}(\mathbf{x}).$$

LMOV conjecture. For every $\mu\in\mathcal{P}_+$, the function $F_{\mu}^{(X,\mathcal{D})}$ has the integral expansion given by the right-hand side of the open multiple-covering formula, namely the expansion in terms of integers $n_{\mu,g,Q}$.

This conjecture predicts the integrality of the open topological-string BPS invariants, refining the multiple-cover formula for open strings and generalizing the closed-string Gopakumar–Vafa integrality structure. The supplied text gives no resolution status.