Erdős Problems 97 and 96: the combined convex-distance mission

The declaration asserts the conjunction of the two preceding targets. First, for every nonempty finite point set $A$ in the Euclidean plane whose points are all absent from the convex hull over $\mathbb{R}$ of the other points, at least one $p\in A$ has fewer than four points of $A$ at every positive exact distance from $p$. Second, there are $C\ge0$ and $N$ such that for every natural $n\ge N$, the supremum of the numbers of unordered unit-distance pairs among convexly independent $n$-point sets, cast to $\mathbb{R}$, is at most $Cn$; the supremum is the natural-number supremum of the set of counts and uses its empty-set convention. The theorem body is sorry, so this conjunction is an asserted target rather than an established result.