The quantum K-theoretic abelian/non-abelian correspondence conjecture

Let $V//G$ be a GIT quotient satisfying the source's mild hypotheses, including that the $G$-unstable locus has codimension two. Let $T_G$ be a maximal torus of $G$, let $W$ be its Weyl group, and let $QK^{tw}(V//T_G)$ denote the twisted quantum $K$-ring of the abelianization. The twisting is determined by $\prod_rEu_{\lambda_r}(L_r)$, where $Eu_{\lambda_r}$ is the $\mathbb{C}^*$-equivariant Euler class for fiber scaling on $L_r$, with $\lambda_r\lambda_{-r}=1$ and $w\lambda_r=\lambda_{wr}$. The quantum abelian/non-abelian correspondence conjecture. The map $\phi_Q$ is a surjective ring homomorphism

$$\phi_Q:QK^{tw}(V//T_G)^W\twoheadrightarrow QK(V//G),$$

and it preserves quantum pairings:

$$\frac{1}{|W|}((a,b))^{tw}=((\phi_Q(a),\phi_Q(b))).$$

This is proposed as a quantum $K$-theoretic generalization of the classical abelian/non-abelian correspondence; the source uses it to relate flag varieties to their abelianizations.