Kawauchi–Nakanishi conjecture on $t_4$-moves and homotopy

Let $L_1$ and $L_2$ be links. Two links are $t_4$, $\bar t_4$ equivalent if one can be obtained from the other by $t_4$ and $\bar t_4$ moves and isotopy. Kawauchi–Nakanishi conjecture. If $L_1$ and $L_2$ are homotopic, then they are $t_4$, $\bar t_4$ equivalent. In particular, every knot is $t_4$, $\bar t_4$ equivalent to the unknot. The conjecture is disproved for links with three or more components; for two-component links, the corresponding classification by $T_2$ or the Hopf link remains open.