Kusner–McGrath's eigenspace conjecture for embedded free boundary minimal surfaces

Let $S$ be a properly embedded free boundary minimal surface in the unit ball $\mathbb B^n$. Consider the first Steklov eigenspace of $S$, and let the coordinate functions of the embedding be viewed as functions on $S$.

Kusner–McGrath's conjecture. The first Steklov eigenspace of $S$ coincides with the span of the coordinate functions of the embedding.

This is presented as a stronger version of the Fraser–Li spectral-index conjecture. The supplied material reports verification under substantially weaker symmetry assumptions, but no general resolution.