Finitely generated pants-like COL2 groups and hyperbolic 3-manifold groups

Let $G$ be a finitely generated torsion-free discrete subgroup of $\operatorname{Homeo_+}(S^1)$. A pants-like $\operatorname{COL}_2$ group has a pants-like collection of two invariant laminations, and “virtually” means after passing to a subgroup of finite index; the laminations are required to be loose. The pants-like COL2–3-manifold group conjecture. $G$ is virtually a pants-like $\operatorname{COL}_2$ group with loose laminations if and only if $G$ is virtually a hyperbolic $3$-manifold group. One implication is explained using Agol’s Virtual Fibering theorem and stable and unstable laminations of pseudo-Anosov maps; the converse is presented as requiring, possibly together with Cannon’s conjecture, a convergence-group argument.