The distance-sum characterization of cross-polytopes

Let $A_1,\ldots,A_{2n}$ be different points in $\mathbb{R}^n$, and let $\Gamma$ be a sphere with $\dim\Gamma=n-1$. Suppose that

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

is independent of the position of $M$ on $\Gamma$ for $\lambda=2,4,6$.

Cross-polytope conjecture. Then $A_1,\ldots,A_{2n}$ are the vertices of an $n$-dimensional cross-polytope.

The surrounding argument establishes related extremal results by induction, but the supplied text does not establish this exact assertion.