Structural classification of minimally angle-rich point sets

Classify, up to the stated notion of essential equivalence, all finite non-collinear planar point sets P for which the number of distinct angles determined by triples of points satisfies the sharp linear-order bound |\mathcal{A}(P)|\gg |P|, and determine whether every such example essentially arises from either the perpendicular-bisector construction or a regular polygon together with its centre.

Background

The paper discusses the structural problem associated with the sharp lower bound for the number of distinct angles determined by a finite non-collinear point set in the plane. It identifies two known constructions attaining linear-order growth: points arranged on a perpendicular bisector so that the resulting angles form an arithmetic progression, and the vertices of a regular polygon together with its centre. The cited conjecture asks whether these constructions account, essentially, for all extremal examples.