Abelian/non-Abelian correspondence for twisted quantum K-rings

Let $G$ be a reductive group acting on a vector space, let $T$ be a maximal torus with Weyl group $W$, and write $Y=V//T$ and $X=V//G$. Let $L_r$ be the bundle on $Y$ associated to a root $r$, and let $QK^{tw}(Y)$ denote the quantum $K$-ring twisted by $Eu_\lambda(\bigoplus_r L_r)$. Let $\phi_Q$ be the specialization map on $W$-invariants, extended to Novikov variables and specialized at $\lambda=1$. Abelian/non-Abelian correspondence for quantum $K$-rings. The map $\phi_Q$ is a surjective ring homomorphism

$$\phi_Q:QK^{tw}(Y)^W\twoheadrightarrow QK^*(X).$$

This is the main quantum $K$-ring formulation of the abelian/non-Abelian correspondence, extending the classical correspondence and relating the twisted theory of the torus quotient to the ordinary theory of the reductive-group quotient. The source provides no evidence that the conjecture has been resolved.