Kawauchi's 4-move conjecture for link-homotopic links

Let two links be link-homotopic if one can be obtained from the other by finitely many self-crossing changes. Two links are 4-equivalent if one can be reached from the other by finitely many 4-moves, their inverses, and Reidemeister moves. Kawauchi's 4-move conjecture. If two links are link-homotopic, then they are 4-equivalent. In particular, every knot is 4-equivalent to the unknot. The conjecture is known for 2-bridge links, pretzel links, and closed 3-braids, but the general statement remains open.