The genus-zero multiple-cover formula for local snc del Pezzo surfaces

Let $S$ be a rank $2$ local snc del Pezzo surface, let $D$ be a polarization, and let $N^0_\gamma$ and $n^{0,D}_\gamma$ denote its local genus-zero Gromov–Witten and local genus-zero BPS invariants. For a curve class $\gamma$, write $k\mid\gamma$ when $\gamma$ is divisible by $k$, with quotient class $\gamma/k$. Local multiple-cover conjecture.

$$N^0_\gamma = \sum_{k\mid \gamma}\frac{n^{0,D}_{\gamma/k}}{k^3}.$$

This is the expected genus-zero relationship between Gromov–Witten and BPS invariants in the local setting. The authors check it in two simple examples with primitive curve classes, but the general statement remains open.