On the universal sl₂ invariant of boundary bottom tangles

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The universal sl₂ invariant of bottom tangles has a universality property for the colored Jones polynomial of links. Habiro conjectured that the universal sl₂ invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra Uₕ(sl₂) of the Lie algebra sl₂. In the present paper, we prove an improved version of Habiro's conjecture. As an application, we prove a divisibility property of the colored Jones polynomial of boundary links.