Relative BPS integrality conjecture for del Pezzo surfaces

Let $S$ be a del Pezzo surface with boundary divisor $E$, let $\beta\in\operatorname{H}_{2}(S,\mathbb Z)$ be an effective curve class, and set $w=\beta\cdot E$. Let $n_{S}[dw]\in\mathbb Q$ be the relative BPS state counts of class $d\beta$, defined for $d\geq 1$ by extracting the multiple-cover contributions from the relative Gromov–Witten invariants.

Relative BPS integrality conjecture. For every $d\geq 1$,

$$n_{S}[dw]\in\mathbb Z.$$

The conjecture asserts integrality of the relative BPS numbers, despite their definition through rational relative Gromov–Witten invariants and multiple-cover contributions. The supplied source does not provide evidence of a resolution, so the conjecture is recorded as open.