The high-dimensional balanced non-group-balanced configuration conjecture

Let a balanced discrete subset $\mathcal{C}\subset\mathbb{R}^n$ be one for which, for every $x\in\mathcal{C}$ and every distance $d$, the set

$$\{y\in\mathcal{C}:|x-y|=d\}$$

either is empty or has centroid $x$. Let $\operatorname{Aut}(\mathcal{C})$ be the rigid motions preserving $\mathcal{C}$. The set is group-balanced if, for every $x\in\mathcal{C}$, its stabilizer in $\operatorname{Aut}(\mathcal{C})$ fixes only $x$.

High-dimensional balanced non-group-balanced configuration conjecture. If $n$ is sufficiently large, then there exists a discrete subset of $\mathbb{R}^n$ that is balanced but not group-balanced.

This conjecture predicts that the planar phenomenon does not persist in all dimensions: sufficiently high-dimensional Euclidean space should contain balanced configurations with additional stabilizer-fixed points. The source gives no resolution or specified threshold for “sufficiently large,” so the conjecture remains open.