Montesinos–Morton branched-cover conjecture for fibered links

Let $L$ be a fibered link in $S^3$. A simple branched cover is a map $\pi\colon S^3\to S^3$ branched over a link $L'$, and $\alpha$ and $\beta$ are unknots that are braid axes of $L'$. Montesinos–Morton's conjecture. For every fibered link $L$, there is a simple branched cover

$$\pi\colon S^3\to S^3$$

branched over a link $L'$ and two unknots $\alpha$ and $\beta$, which are braid axes of $L'$, such that $\pi^{-1}(\alpha)=L$ and $\pi^{-1}(\beta)$ is the unknot. Montesinos and Morton proved this for fibered links obtained from the unknot and its fiber disk by Hopf plumbings without deplumbings; the general assertion remains open.