The genus-zero log-open correspondence for nef Looijenga pairs

Let $(X,D)$ be a nef log Calabi–Yau surface pair with $D=D_1+\cdots+D_l$, and let $(Y,L)$ be the associated toric Calabi–Yau threefold with framed Aganagic–Vafa branes $L_i^{[f_i]}$ for $i<l$. Let $\iota$ be the canonical identification of the relevant curve and relative homology classes, and let $O_{0,\iota(d)}$ denote the corresponding genus-zero open invariant. The genus-zero log-open correspondence.

$$N_{0,d}^{\rm loc}(X,D_1 +\dots + D_l) =O_{0, \iota(d)}(Y,L_1^{[f_1]} \sqcup \dots \sqcup L_{l-1}^{[f_{l-1}]}).$$

The statement identifies local surface invariants with open Gromov–Witten invariants of the associated toric Calabi–Yau geometry. In the source it is presented as the expected general correspondence for $l>1$, so its general validity remains open.