Temperley–Lieb unfaithfulness conjecture for knots with trivial Jones polynomial

Let $B_n$ be a braid group, and let its Temperley–Lieb representation be the representation into the Temperley–Lieb algebra. A knot is non-trivial if it is not isotopic to the unknot. Temperley–Lieb unfaithfulness conjecture. If the Temperley–Lieb representation of the braid group is unfaithful, then there exists a non-trivial knot with Jones polynomial equal to one. This conjecture connects kernels of braid-group representations with the question of whether the Jones polynomial detects the unknot; the source presents it as widely assumed, but gives no proof.