The low BKL-exponent obstruction conjecture for L-space branched covers

Let $L$ be a prime, fibred, strongly quasipositive link. Write $k(L)$ for its BKL-exponent, defined as the maximum integer $k$ such that $L$ is the closure of $\delta_n^kP$ for some $n\geq 2$, $k\geq 0$, and BKL-positive braid $P\in B_n$. Low BKL-exponent obstruction conjecture. If $k(L)\leq 1$, then no cyclic branched cover $\Sigma_n(L)$ is an L-space. Together with the stated characterization of simply laced arborescent links for BKL-exponent at least $2$, this would prove the broader conjectural classification of prime fibred strongly quasipositive links with an L-space cyclic branched cover. The supplied text does not resolve this case.