Characterization of Hilbert spaces by minimal K-sets

Let $\mathbb{X}$ be an $n$-dimensional normed linear space, with unit sphere $S_{\mathbb{X}}$. For subsets $A,B\subseteq\mathbb{X}$, write $A\perp_B B$ when every element of $A$ is Birkhoff-James orthogonal to every element of $B$. A nonempty subset $A\subset S_{\mathbb{X}}$ is a minimal $\mathcal K$-set if it has the minimality property considered in the article.

Hilbert-space characterization conjecture. The following conditions are equivalent: $\mathbb{X}$ is a Hilbert space; and a nonempty set $A\subset S_{\mathbb{X}}$ is a minimal $\mathcal K$-set if and only if

$$A=\{x_1,x_2,\dots,x_n\},$$

where $x_1,x_2,\dots,x_n$ are linearly independent, and whenever $A=A_1\cup A_2$ with $A_1\neq\varnothing$ and $A_2\neq\varnothing$, one has $A_1\not\perp_B A_2$.

The conjecture proposes that this description of minimal $\mathcal K$-sets characterizes finite-dimensional Hilbert spaces among normed linear spaces.