Erdős Problem #97 — Equidistant Points in Convex Polygons

For a finite set $A\subset\mathbb{R}^2$, a point $p$ has at least $n$ equidistant points in $A$ if there exists $r>0$ such that at least $n$ points $q\in A$ satisfy $d(p,q)=r$. Say that $A$ has the $n$-equidistant property if every point $p\in A$ has at least $n$ equidistant points in $A$. Is it true that every nonempty finite set $A\subset\mathbb{R}^2$ in convex position fails to have the $4$-equidistant property; equivalently, does every convex polygon have a vertex with no other four vertices equidistant from it?