The distance-sum characterization of regular simplices

Let $A_1,\ldots,A_{n+1}$ be points in $\mathbb{R}^n$, and let $\Gamma$ be a sphere. Suppose that

$$\sum_{i=1}^{n+1}|MA_i|^{\lambda}$$

is independent of the position of $M$ on $\Gamma$ for $\lambda=2,4$.

Regular simplex conjecture. Then $A_1\ldots A_{n+1}$ are the vertices of a regular simplex.

The statement extends the planar characterization to regular simplices in arbitrary dimension; its status is not resolved in the supplied text.