Open orbifold mirror symmetry conjecture for Gromov–Witten potentials

Let $X$ be a toric Calabi–Yau threefold and $\mathfrak{X}$ its orbifold phase. Let $\{t_i^X\}$ and $\{t_i^{\mathfrak{X}}\}$ denote their quantum parameters, with compact flat coordinates $\hat t_i^X,\hat t_i^{\mathfrak{X}}$ and non-compact coordinates $\tilde t_i^X,\tilde t_i^{\mathfrak{X}}$. Suppose the compact coordinates and prepotentials are related by

$$\begin{pmatrix}\hat t_i^X\\ \partial\mathcal{F}_0^X/\partial\hat t_i^X\end{pmatrix}= \begin{pmatrix}A&B\C&D\end{pmatrix} \begin{pmatrix}\hat t_i^{\mathfrak{X}}\\ \partial\mathcal{F}_0^{\mathfrak{X}}/\partial\hat t_i^{\mathfrak{X}}\end{pmatrix},$$

and the open parameters satisfy $x_j=w_j\prod_kq_k^{r_k}$. Let $W_h^{(g)}$ be the open correlators of $X$, with $W_1^{(0)}=d\lambda_X$ when $(g,h)=(0,1)$, and let $M=\begin{pmatrix}A&B\C&D\end{pmatrix}$ be the corresponding change-of-basis matrix. Define the transformed correlators $\mathcal{W}_h^{(g)}$ as in the source for $2g+h>1$, and set $\widetilde W_1^{(0)}=W_1^{(0)}$.

Open orbifold mirror symmetry conjecture. The open orbifold potentials $F_{g,h}^{(\mathfrak{X},\mathcal{L})}$ are given, after applying the orbifold open and closed mirror maps, by

$$\int\mathcal{W}_h^{(g)}(p_1,\ldots,p_h)\frac{dp_1}{p_1}\dots\frac{dp_h}{p_h}.$$

The conjecture proposes that the symplectic change-of-basis data relating the large-radius and orbifold points suffice to reconstruct open Gromov–Witten potentials of the orbifold from those of $X$. The source gives no resolution status, and the transformation defining $\mathcal{W}_h^{(g)}$ is referenced but not reproduced here.