Critical catenoid uniqueness for first-Steklov-eigenfunction embeddings

Let $N$ be an orientable surface with two boundary components, and let $\Psi:(N,g)\to\mathbb{B}^3$ be a free boundary minimal embedding by $\sigma_1(N,g)$-eigenfunctions, with multiplicity of $\sigma_1(N,g)$ equal to $3$. Suppose $\Xi$ is a free boundary harmonic map to $\mathbb{B}^3$ by $\sigma_1(N,g)$-eigenfunctions and is not $A\Psi$ for any $A\in O(3)$. First-Steklov uniqueness conjecture. Then $(N,g)$ is the critical catenoid. The source presents this as an open conjecture that would simplify the proof of the existence theorem.