The square-root growth conjecture for distinct distance subsets

Let $P$ be a set of $N$ points in the plane, and let $\delta(N)$ denote the largest size of a subset of $P$ whose pairwise distances are all distinct. For every $\epsilon>0$, consider a constant $c_\epsilon>0$ independent of $N$. Distinct distance subset conjecture. For every $\epsilon>0$, there exists some constant $c_\epsilon>0$ such that

$$\delta(N)\ge c_\epsilon N^{1/2-\epsilon}.$$

The preceding argument gives a lower bound of order $N^{1/3}/\log N$, while the conjectured bound is close to the square-root upper bound established earlier in the paper. The source presents this as an inclination to conjecture, so its resolution is not supplied here.