On the universal sl2 invariant of boundary bottom tangles

The universal sl2 invariant of bottom tangles has a universality property for the colored Jones polynomial of links. A bottom tangle is called boundary if its components admit mutually disjoint Seifert surfaces.
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). In this talk, we show an improvement of Habiro's conjecture.