Equivalence of Fukaya categories via the moment Lagrangian correspondence

Prove that the Lagrangian correspondence µ^{-1}(0) ⊂ X × X//G, where X is a compact symplectic manifold equipped with a Hamiltonian action of a compact connected Lie group G and with anticanonical linearization L = det(TX) (so that the Fukaya categories are defined in the monotone A-model setting), induces an A∞-equivalence between the Fukaya category F(X) and the Fukaya category F(X//G).

Background

The paper studies a two-dimensional A-model analogue of the principle that quantization commutes with reduction, relating the gauged A-model of a Hamiltonian G-manifold X to the A-model of its symplectic reduction X//G.

Within this framework, the authors formulate a categorical strengthening: the Lagrangian correspondence given by the zero level of the moment map should induce an equivalence of Fukaya categories. This conjecture sits in the monotone setup where L = det(TX), ensuring the Fukaya categories F(X) and F(X//G) are defined.