The asymptotic two-smallest-distances conjecture for planar point sets

Let $S$ be a set of $n$ points in the plane. Let $m_1(S)$ and $m_2(S)$ be the numbers of occurrences of the smallest and second smallest distances in $S$, respectively, and let $f(n)$ be the maximum of $m_1(S)+m_2(S)$ over all such sets. For the triangular lattice, let $e(n)$ denote the maximum number of edges in an induced $n$-vertex subgraph of the specified triangular-lattice Cayley graph; equivalently, for $n\geq 3$,

$$e(n):=\begin{cases} 6n-4\sqrt{6n-6}&\text{if }n=24k^2-24k+7\text{ for some }k\in\mathbb{N},\\ \lfloor 6n-\sqrt{96n-63}\rfloor&\text{otherwise}.\end{cases}$$

Asymptotic two-smallest-distances conjecture. For any sufficiently large $n$, $f(n)=e(n)$, and the only sets $S$ attaining $f(n)=m_1(S)+m_2(S)$ are geometrically similar to the extremal sets on the triangular lattice. The theorem preceding this conjecture supplies the lower bound $f(n)\geq e(n)$, while the known general upper bound is $f(n)\leq 6n$. The conjecture asserts both eventual sharpness of the lattice construction and uniqueness up to geometric similarity.