Milnor-invariant criterion for the universal sl_2 invariant of bottom tangles

Let $T$ be an $n$-component bottom tangle with $0$-framing, and suppose that all Milnor $\bar\mu$ invariants of $T$ vanish. Let $J_T$ denote its universal $sl_2$ invariant. The relevant completed tensor product is $(\bar U_q^{\operatorname{ev}})\,\hat{}^{\;\hat \otimes n}$.

Milnor-invariant conjecture. One should have

$$J_T\in (\bar U_q^{\operatorname{ev}})\,\hat{}^{\;\hat \otimes n}.$$

The paper presents this as a natural expectation because all Milnor $\bar\mu$ invariants vanish for ribbon and boundary bottom tangles. The claim is not established in the supplied text, although it would extend the proved result for boundary bottom tangles.