Fraser–Li conjecture on the critical catenoid

Let $B^3\subset\mathbb{R}^3$ be the unit ball, and let $\Sigma_{\mathrm{cc}}$ denote the critical catenoid, the free boundary minimal annulus obtained by scaling the Euclidean catenoid to meet $\partial B^3$ orthogonally. Fraser–Li conjecture. Up to ambient congruence, the critical catenoid is the unique properly embedded free boundary minimal annulus in $B^3$. This is the free boundary analogue of Lawson-type uniqueness for the Clifford torus. The conjecture remains open in full generality.