How large can the difference between the largest and second-largest distance multiplicities be among $n$ planar points?

A superlinear lower bound: there is a $c>0$ with $$\text{extremalGap}(n)\geq n^{1 + c/\log\log n}$$ for all large $n$, so $\max (f(d_1)-f(d_2))$ grows faster than any linear function of $n$. Determining the exact order remains open, which is erdos_959.