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This paper is the $K$-theoretic analogue of a recent new proof, given by thefirst named author, of Peterson-Lam-Shimozono's theorem via Savelyev'sgeneralization of Seidel representations. The outcome is a new proof ofLam-Li-Mihalcea-Shimozono's conjecture, including its extension to theparabolic case, which was first verified by Kato.

Quantum $K$-theory of $G/P$ and $K$-homology of affine Grassmannian