Is the following conjecture true:
If $G$ is a group of rotations of the sphere and $G$ contains two noncommuting rotations of infinite order, then $G$ has a free subgroup of rank $2$.
By the Tits alternative, it would suffice to show that $G$ is not almost solvable (i.e., has no solvable subgroup of finite index). If true, this would imply many specific cases, going back to Hausdorff, of free groups of rotations.