Global validity of the regular-polygon maximizer property

Let $N$ be the number of points and let $(P_m)$ denote the property that the maximizer represented by regular polygons has the specified geometric extremal behavior for $m$ points. Global maximizer property. The property $(P_m)$ holds globally for any $m=2,\dots, [\frac{1}{2}N]$. The global nature of the regular-polygon maximizer is reduced to this geometric claim; the preceding argument establishes the corresponding local result, while the global validity is presented as the more difficult unresolved question.