Fraser-Li conjectured that any FBMS is embedded by its first eigenfunctions. This would impose a strong restriction on the conformal classes of surfaces with boundary that admit such a minimal embedding. We prove the conjecture in the genus-zero case. I will explain how this implies the uniqueness of the critical catenoid. These results can be regarded as free-boundary analogues of Yau’s conjecture on the first eigenvalue of minimal hypersurfaces in spheres and the Lawson conjecture on the uniqueness of the Clifford torus. Joint work with O. Chodosh.