The extremal-distance characterization of regular polygons

Let $A_1,A_2,\ldots,A_n$ be different points in the plane and let $\Gamma$ be a circle. Suppose that

$$\sum_{i=1}^n |PA_i|^{2n-2}$$

is constant for $P\in\Gamma$.

Regular polygon conjecture. Then $A_1,A_2,\ldots,A_n$ are the vertices of a regular polygon inscribed in a circle concentric to $\Gamma$.

The preceding theorem proves the analogous assertion when the sums for all exponents $2,4,\ldots,2n-2$ are constant; the conjecture asks whether the single highest even exponent already suffices.