Reid's conjecture on conjugate generators of knot groups

Let $K$ be a knot, and let $\pi_1(S^3\smallsetminus K)$ denote its knot group. Suppose this group is generated by two conjugate elements. Reid's conjecture. The elements are peripheral, and hence the knot is two-bridge. The conjecture is motivated by the characterization of two-bridge knot and link groups generated by two peripheral elements. The statement has recently been proved, so it is no longer open.