Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where $q \in \mathbb{C}^\times$ is not a root of unity.

Lusztig showed that the braid group $B_{\mathfrak{g}}$ (an infinite cover of the Weyl group of $\mathfrak{g}$ obtained by dropping the relations that the generators are involutive) acts by algebra automorphisms on $U_q(\mathfrak{g})$. (In fact there are several different ways that $B_{\mathfrak{g}}$ can act, but they are closely related and it doesn't matter which one we pick for the purposes of this question.)

Has anybody seen anything addressing the question of whether this action is faithful? I would much appreciate an explanation or a reference that resolves the question either way. Thank you!