Quantum GIT conjecture for Fukaya categories

Let $X$ be a monotone almost complex manifold with a Hamiltonian action of a group $G$, moment map $\mu$, and symplectic quotient $X/\!/G=\mu^{-1}(0)/G$. Let $\mathcal{F}(X)$ denote its Fukaya category, and let $\mathcal{F}(X)^G$ denote the category of $G$-equivariant objects. The Lagrangian correspondence

$$\mu^{-1}(0)\subset X\times X/\!/G$$

induces an equivalence of categories

$$\mathcal{F}(X)^G\equiv\mathcal{F}(X/\!/G).$$

This is the Fukaya-category, or $A$-model, form of the quantum GIT conjecture: quantization should commute with symplectic reduction under the stated positivity assumptions. The paper presents this as an expected equivalence in the two-dimensional setting; its general validity remains open.