The polar-duality extremum conjecture for regular polytopes

Let $P$ be an $n$-dimensional regular polytope in $\mathbb{R}^n$, inscribed in the unit sphere $U$. Let $S$ be the polytope obtained by polar reciprocation of $P$ in $U$, and let $P'$ be a homothetic copy of $S$ inscribed in $U$. Write $\mathbf{V}$ and $\mathbf{V'}$ for the vertex sets of $P$ and $P'$, respectively. For a sphere concentric with $U$, consider

$$\sum_{A\in\mathbf{V}}|MA|^{\lambda}$$

and

$$\sum_{A\in\mathbf{V'}}|MA|^{\lambda}.$$

Polar-duality conjecture. If the first sum attains its maximum at a point $N$, then the second sum attains its minimum at $N$, and conversely. If the first sum is independent of the movement of $M$ on $\Gamma$, then the second sum is also independent of that movement, and conversely.

The claim proposes a common extremal and constancy relationship between polar-dual regular polytopes, motivated by the established cube/cross-polytope and simplex cases; the supplied text does not state a general proof.