Uniqueness conjecture for critical rotational annuli in spherical and hyperbolic balls

Let $\mathbb{S}^3_+$ denote the round hemisphere and let $\mathbb{H}^3$ denote hyperbolic three-space. A free boundary minimal annulus is a compact minimal annulus in a geodesic ball whose boundary lies on the boundary sphere and which meets that sphere orthogonally. The critical rotational annuli are the only embedded free boundary minimal annuli in geodesic balls of $\mathbb{S}^3_+$ and $\mathbb{H}^3$. Families of immersed and non-rotational free boundary minimal annuli are known, so the conjecture concerns uniqueness among embedded annuli; the general assertion remains open.