Does every convex polygon have a vertex with no other 4 vertices equidistant from it?

Fishburn and Reeds [FiRe92] also proved that the smallest $n$ for which there exists a convex $n$-gon and a cut $\{A, B\}$ of its vertices such that $|\{b \in B : d(a, b) = 1\}| ≥ 3$ for all $a \in A$, and $|\{a \in A : d(a, b) = 1\}| ≥ 3$ for all $b \in B$, is $n = 20$.