Rotational symmetry conjecture for free boundary minimal annuli in rescaled spherical caps

Let $R\in[0,\frac{\pi}{2}]$ and let $\mathbb{M}^3_{\kappa_R}$ be the space form of curvature $\kappa_R=\sin^2 R$. An embedded free boundary minimal annulus in $B_{R\csc R}\subset\mathbb{M}^3_{\kappa_R}$ is an embedded minimal annulus whose boundary lies on $\partial B_{R\csc R}$ and meets it orthogonally. The rotational symmetry conjecture. If

$$(\Sigma,\partial\Sigma)\hookrightarrow(B_{R\csc R},\partial B_{R\csc R})$$

is an embedded free boundary minimal annulus, then $\Sigma$ is rotationally symmetric. This is a natural analogue of the Lawson and Nitsche–Fraser–Li uniqueness conjectures. The statement is presented as an open rigidity problem; no resolution is supplied in the source.