The tight-pair global fixed-point conjecture

Let $(\operatorname{\mathcal{L}},G)$ be a tight pair, where $G$ acts on the circle preserving the lamination system $\operatorname{\mathcal{L}}$. Tight-pair global fixed-point conjecture. The group $G$ admits no global fixed point; therefore, it contains a nonabelian free subgroup. This is proposed as a future direction, and the source notes that the claim is known for sticky pairs but appears more difficult for slippery pairs.