Eventually more than $A\,n$ distinct distances in general position for every $A\le 1/3+1/130$

Let $A$ be a real number with $A\le 1/3+1/130$. Then there is a threshold $N$ such that the following holds for every $n\ge N$. Let $p=(p_1,\dots,p_n)$ be an injective configuration of $n$ points in the plane whose image $P=\{p_1,\dots,p_n\}$ has no three collinear points and no four points on a common circle, and let $D(P)$ be the number of distinct distances between distinct points of $P$. Then

The threshold may depend on $A$, and the conclusion holds for every qualifying configuration of size $n$. This is the eventual, fixed-coefficient form of the bound $D(P)>(1/3+1/130)n$ for $n\ge 10000$. Erdős Problem 98 asks for this conclusion for every real $A>0$; the statement here covers the coefficients $A\le 1/3+1/130$ only.

Formalization Note The threshold is expressed with Filter.atTop on the natural numbers, and $D(P)$ is NumDistances p.