It has been known since their birth in Bao and Wang’s work that the
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\imath
canonical bases of
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\imath
quantum groups are not stable in general. In the author’s previous work, the stability of
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\imath
canonical bases of certain quasi-split types turned out to be closely related to the theory of
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crystals. In this paper, we prove the stability of
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canonical bases of irreducible finite type of real rank
1
1
, for which the theory of
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crystals has not been developed, by means of global and local crystal bases.