Point-separation conjecture for first Steklov eigenfunction maps

Let $N^2\subset\mathbb{B}^n$ be a free boundary branched minimal immersion by first Steklov eigenfunctions, with $\partial N$ having at least two components. Steklov point-separation conjecture. There exists a pair of points $p,q$ in distinct components of $\partial N$ such that every map

$$F:(N,\partial N)\to(\mathbb{B}^n,\mathbb{S}^{n-1})$$

by first Steklov eigenfunctions satisfies $F(p)\neq F(q)$. This is proposed as one of the remaining ingredients for the existence theory of $\bar{\sigma}_1$-maximizing metrics; the source gives no resolution.