The branched-cover L-space conjecture for prime strongly quasipositive links

Let $L$ be a link in $S^3$, and for each integer $n\geq 2$ let $\Sigma_n(L)$ denote the canonical $n$-fold cyclic cover branched over $L$. A link is simply laced arborescent if it is the boundary of an oriented surface obtained by plumbing positive Hopf bands according to one of the simply laced Dynkin diagrams $A_m$ ($m\geq 1$), $D_m$ ($m\geq 4$), $E_6$, $E_7$, or $E_8$. The branched-cover L-space conjecture. If $L$ is a prime, fibred, strongly quasipositive link for which some $\Sigma_n(L)$ is an L-space, then $L$ is simply laced arborescent. This would characterize the prime fibred strongly quasipositive links having an L-space cyclic branched cover; the converse is supplied by the simply laced arborescent examples, whose double branched covers are L-spaces. The conjecture is not resolved in the supplied text.