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