The higher genus log/open correspondence for quasi-tame Looijenga pairs

Let $Y(D)$ be a quasi-tame Looijenga pair with components $D_1,\ldots,D_l$, let $Y^{\rm op}(D)$ be its associated open geometry, and let $d$ be a curve class. Write $[n]_q=q^{n/2}-q^{-n/2}$, set $q=\mathrm{e}^{\mathrm{i}\hbar}$, and let $\mathbb{O}_d$, $\mathbb{N}^{\rm \log}_d$, and $\mathbb{LMOV}_d$ denote the all-genus open, logarithmic, and LMOV generating functions and invariants.

The higher genus log/open correspondence. One has

$$\mathbb{O}_d(Y^{\rm op}(D))(\hbar)=\left(\prod_{j=1}^{l-1}\frac{(-1)^{d\cdot D_j-1}}{d\cdot D_j}\right)\frac{(-1)^{d\cdot D_l-1}}{[d\cdot D_l]_q}\mathbb{N}^{\rm \log}_d(Y(D))(\hbar).$$

Moreover,

$$\mathbb{LMOV}_d(Y^{\rm op}(D))(\hbar)=[1]_q^2\left(\prod_{i=1}^l\frac{1}{[d\cdot D_i]_q}\right)\sum_{k\mid d}\frac{(-1)^{d/k\cdot D+l}\mu(k)}{[k]_q^{2-l}k^{2-l}}\mathbb{N}^{\rm \log}_{d/k}(Y(D))(k\hbar)\in\mathbb{Z}[q,q^{-1}].$$

This all-genus extension relates logarithmic and open Gromov--Witten theories through a $q$-hypergeometric identity. The source records that two non-tame quasi-tame cases were subsequently proved, while the full correspondence remains unresolved.