The self-similarity conjecture for extremal angle configurations

A point configuration $\mathcal P$ is self-similar if there is a point $A\in\mathcal P$ such that every angle formed by three points of $\mathcal P$ can also be formed with $A$ as one of the points: for all $B,C,D\in\mathcal P$, there exist $E,F\in\mathcal P$ such that $\angle BCD=\angle AEF$ or $\angle BCD=\angle EAF$. Self-similarity conjecture. For $n$ sufficiently large, a configuration of $n$ points in general position with the smallest possible number of distinct angles exhibits self-similarity. The paper notes that self-similar configurations have at most $3\binom{n-1}{2}=O(n^2)$ distinct angles and presents constructions with this property, but does not establish that every extremal configuration has it.