The K-Peterson conjecture for equivariant quantum K-theory

Let $G$) be a simple and simply-connected complex Lie group with Borel subgroup $B$, maximal torus $T$, Weyl group $W$, affine Weyl group $W_{\mathrm{af}}=W\ltimes Q^\vee$, and coroot lattice $Q^\vee$. Let $\Lambda$ be the weight lattice, and write $R(T)=K_T(\operatorname{pt})$. Let $\operatorname{QK}_T(G/B)$ have Schubert basis $\mathcal{O}^w$ for $w\in W$, and let $K_0^T(\operatorname{Gr})$ have Schubert basis $\mathcal{O}_x$ indexed by $x\in W_{\mathrm{af}}^-$. For $ut_\lambda,vt_\mu,wt_\nu\in W_{\mathrm{af}}^-$ and $\eta\in Q^\vee$, assume that $\nu=\lambda+\mu$. The K-Peterson conjecture. The structure constants satisfy

$$c_{ut_\lambda,vt_\mu}^{wt_{\nu+\eta}}=N_{u,v}^{w,\eta}$$

in $K_T^*(\operatorname{pt})$, where the $c$'s are the structure constants of $K_0^T(\operatorname{Gr})$ with respect to the classes $\mathcal{O}_x$, and the $N$'s are the structure constants of $\operatorname{QK}_T(G/B)$ with respect to the classes $\mathcal{O}^w$. This conjecture predicts the coincidence of the Schubert structure constants for the Pontryagin product in affine-Grassmannian $K$-homology and equivariant quantum $K$-theory of the flag variety. It would imply finiteness of quantum $K$-theory multiplication and leads to an isomorphism after localizing the relevant affine-Grassmannian and quantum parameters; the paper verifies it in several situations, but it is not established in general.