Erdős Problem #958 — Configurations with consecutive distance multiplicities

For a finite set $A\subset\mathbb R^2$, let $n=|A|$, let $D(A)$ be the set of distances determined by unordered pairs of distinct points of $A$, and let the multiplicity of a distance $d$ be the number of such pairs at distance $d$. Is the following assertion false? For every $n\in\mathbb N$ and every finite set $A\subset\mathbb R^2$ with $|A|=n$, if $|D(A)|=n-1$ and the set of distance multiplicities is $\{1,2,\ldots,n-1\}$, then $A$ consists either of equally spaced points on a line or of points on a circle with equally spaced angular increments. More explicitly, the line condition is that there exist $a,v\in\mathbb R^2$ with $v\ne0$ such that $A=\{a+i v: i\in\mathbb N,\ i<n\}$, and the circle condition is that there exist $c\in\mathbb R^2$ and $r,\theta,\alpha\in\mathbb R$ with $r>0$ and $\alpha\ne0$ such that $A=\{c+r(\cos(\theta+i\alpha),\sin(\theta+i\alpha)):i\in\mathbb N,\ i<n\}$.