Bonatti's splitting conjecture for minimal circle actions

Let $G\leq \operatorname{Homeo}_+(S^1)$ be a subgroup such that every non-trivial element has at most two fixed points, and whose action on $S^1$ is minimal. Assume that the action of $G$ is not semi-conjugate to any subgroup of $\operatorname{PSL}_2(\mathbb{R})\leq \operatorname{Homeo}_+(S^1)$. Bonatti's conjecture. Then $G$ splits as an amalgamated product over an abelian subgroup. This conjecture concerns the structure of groups with hyperbolic-like circle dynamics; the supplied text does not state whether it has been resolved.