Descent: a minimal counterexample with more than nine vertices has a removable vertex

For every finite set $A$ of points in the real Euclidean plane, suppose that $A$ is nonempty, every point $a\in A$ is outside the real convex hull of $A\setminus\{a\}$, every point $p\in A$ has some positive real radius $r>0$ for which $\{q\in A : \operatorname{dist}(p,q)=r\}$ has cardinality at least $4$, and $9<|A|$. Suppose also that for every finite set $B$ of points in the plane, whenever $|B|<|A|$, $B$ is nonempty, every point of $B$ is outside the real convex hull of the other points of $B$, and every point of $B$ has a positive radius giving at least $4$ points of $B$ at that distance, the proposition $\mathrm{False}$ holds; this minimality clause quantifies over all such smaller finite sets $B$ and imposes no subset relation between $B$ and $A$. Then there exists a point $x\in A$ such that, after removing $x$ from $A$, every point $p\in A\setminus\{x\}$ has some positive real radius $r>0$ (possibly depending on $p$) with at least $4$ points $q\in A\setminus\{x\}$ at distance $\operatorname{dist}(p,q)=r$.