The Abelian/non-Abelian correspondence for Givental cones

Let $G$ be a reductive group acting on a smooth projective variety $A$, let $T$ be a maximal torus of $G$, and let $W$ be the Weyl group. Write $A /\!/ G$ and $A /\!/ T$ for the corresponding GIT quotients, and let $\operatorname{\mathcal{L}}_{\operatorname{{GM}}} \subset \operatorname{\mathcal{H}}_{A /\!/ G}$ be the Givental--Martin cone obtained as the non-equivariant limit of the Weyl-invariant twisted theory of $A /\!/ T$. The Abelian/non-Abelian correspondence. The Givental--Martin cone equals the Givental cone of the non-Abelian quotient:

$$\operatorname{\mathcal{L}}_{\operatorname{{GM}}} = \operatorname{\mathcal{L}}_{A /\!/ G}.$$

This is a reformulation of the Abelian/non-Abelian correspondence for genus-zero Gromov--Witten invariants, previously conjectured by Ciocan-Fontanine, Kim, and Sabbah. It predicts that the Weyl-invariant twisted theory of the Abelian quotient recovers the Gromov--Witten theory of the non-Abelian quotient.