The two-site mutation conjecture for rational subtangle replacements

Let $\tau$ be a tangle, and let $\alpha_0$ and $\alpha_1$ be sites for a rational subtangle replacement (RSR) on $\tau$. Suppose their lifts to the branched double cover of $\tau$ are isotopic knots. A rational subtangle replacement $\rho\mapsto\rho'$ on either site produces tangles equivalent by a sequence of mutations. Two-site mutation conjecture. There must then be a sequence of mutations of subtangles of $\tau$ taking $\alpha_0$ to $\alpha_1$. The isotopy hypothesis on the lifted knots is necessary, and the conjecture is known in some cases, including when $\tau$ is the unknot and the replacement produces a 2-bridge knot or link; the general assertion remains open.