The open crepant resolution conjecture for toric orbifolds

Let $X$ be a toric variety with at worst Gorenstein quotient singularities. Let $\mathcal{X}$ be the canonical toric orbifold with coarse moduli space $X$, and let $Y$ be a toric crepant resolution of $X$. Denote the flat coordinates on their Kähler moduli spaces by $q$ and $Q$, respectively, and let $l$ be the common dimension of these moduli spaces. Write $W^{LF}_{\mathcal{X}}(q)$ and $W^{LF}_Y(Q)$ for their Lagrangian Floer superpotentials. Open crepant resolution conjecture. There exist $\epsilon>0$, a holomorphic coordinate change $Q(q):(\Delta(\epsilon)-\mathbb{R}_{\leq 0})^l\to(\mathbb{C}^{\times})^l$, and a choice of analytic continuation of the coefficients of $W^{LF}_Y(Q)$ to the target of this map, such that $W^{LF}_Y(Q(q))$ is a holomorphic family of Laurent series near $q=0$ and

$$W^{LF}_{\mathcal{X}}(q)=W^{LF}_Y(Q(q)).$$

This is an open-string version of the crepant resolution conjecture, predicting equality of the generating functions of open Gromov–Witten invariants for a Gorenstein toric orbifold and its toric crepant resolution after analytic continuation and a change of variables.