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For a positive braid group element $b\\in \\operatorname{Br}^+$, Kashiwara--Kim--Oh--Park introduced a subalgebra $\\widehat{\\mathcal A}(b)$ and conjectured that it admits a quantum cluster algebra structure whose cluster monomials belong to the global basis.\n  In this paper, we analyze Lusztig parametrizations of the global basis of $\\widehat{\\mathcal A}(b)$ and study their transition maps under braid moves. 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