The toroidal link conjecture for cyclic branched covers

Let $L$ be a prime toroidal link, and let $cSigma_{\psi}(L)$ denote a cyclic branched cover of $L$. Write $NLS$ for the property of not being an $L$-space, $LO$ for having left-orderable fundamental group, and $CTF$ for supporting a co-orientable taut foliation.

Toroidal link conjecture. All cyclic branched covers $cSigma_{\psi}(L)$ of $L$ are $NLS$, $LO$, and $CTF$.

This is proposed as a broader toroidal case related to an earlier conjecture for prime satellite knots, where the conclusion was expected only for sufficiently large covering degree. Its resolution is not given in the source.