Algebro-geometric proof of the quantum GIT formality isomorphism

Develop an algebro-geometric proof of the isomorphism QH^*(X//G) ≅ QH_G^*(X) ⊗_{H^*(ΩG)} H^*(BG) (with strict tensor product and no higher Tor terms) asserted for compact G-monotone Hamiltonian G-manifolds X with anticanonical linearization.

Background

A central result described in the paper identifies the quantum cohomology of the symplectic reduction X//G with a base change of the LG-equivariant quantum cohomology of X along the unit section of the Toda space, giving a ‘formality’ statement (Formula (0.2)).

While the authors’ approach is Floer-theoretic and TQFT-based, they explicitly note the absence of a known algebro-geometric proof of this formality isomorphism.