Faithfulness of the Jones (Reshetikhin–Turaev) representation

Determine whether the fundamental Reshetikhin–Turaev (Jones) representation of the braid group on four strands is faithful; that is, prove or disprove that the Jones representation distinguishes all elements of the four-strand braid group.

Background

The Jones representation is the fundamental Reshetikhin–Turaev representation underlying the Jones polynomial via quantum group methods. Bigelow showed a deep connection between unfaithfulness of this representation and the existence of knots with trivial HOMFLY-PT (and thus Jones) polynomials.

Since the faithfulness questions for the Burau and Jones representations are linked for four-strand braids, resolving faithfulness for the Jones representation would directly impact the Jones unknot problem and the possibility of constructing explicit counterexamples.