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

Let $L$ be a prime, fibred, strongly quasipositive link. For an integer $n\geq 2$, let $\Sigma_n(L)$ denote the $n$-fold branched cover of $S^3$ branched along $L$, and call $L$ simply laced arborescent if it is the boundary of a surface obtained by plumbing positive Hopf bands according to a simply laced Dynkin tree $A_m$, $D_m$, $E_6$, $E_7$, or $E_8$. An L-space is a rational homology $3$-sphere whose Heegaard Floer homology has minimal possible rank.

The branched-cover L-space conjecture. If some $\Sigma_n(L)$ is an L-space, then $L$ is simply laced arborescent. If $L=K$ is a knot, then $K$ is a $(2,k)$, $(3,4)$, or $(3,5)$ torus knot.

The conjecture generalizes the preceding classification for positive braid, divide, and certain fibred strongly quasipositive knots. Its general case is presented as open in the source.