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

Let $(X,D)$ be a nef Looijenga pair admitting a deformation to a pair $(X',D')$ with $X'$ toric, $D'_i$ prime toric divisors for $i<l$, and $D'_l$ nef. Let $(Y,L)$ be the corresponding Aganagic–Vafa pair, let $\mathbb{N}_{d}^{\rm \log}$ and $\mathbb{O}_{\iota(d)}$ denote the all-genus generating functions for the log and open invariants, and set $q=e^{\mathrm{i}\hbar}$. The higher-genus log-open correspondence.

$$\mathbb{N}_{d}^{\rm \log}(X,D_1 +\dots + D_l) = \left(\prod_{i<l} (-1)^{d \cdot D_i +1} d\cdot D_i\right) (-1)^{d\cdot D_l+1} [d \cdot D_l]_q \,\mathbb{O}_{\iota(d)}(Y,L_1^{[f_1]} \sqcup \dots \sqcup L_{l-1}^{[f_{l-1}]}),$$

where $q=e^{\mathrm{i}\hbar}$.

This conjecture refines the genus-zero log-open correspondence to all genera, replacing the final degree factor by its $q$-deformation. The source presents it as a conjectural formula for nef Looijenga pairs; its general status is open.