Jones polynomial unknot detection

Determine whether the Jones polynomial detects the unknot; specifically, prove or refute that J(K)=1 if and only if K is the unknot.

Background

The Jones polynomial is a powerful knot invariant, but it is unknown whether it uniquely identifies the unknot. In contrast, Khovanov homology is known to detect the unknot (Kronheimer–Mrowka). Establishing whether the Jones polynomial also detects the unknot would clarify the relative strength of these invariants for the unknotting problem.