OPEN

Is there a set of $n$ points in $\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is
\[\ll \frac{n}{\sqrt{\log n}}?\]

Erdős believed this should be possible, and should imply effective upper bounds for [658] (presumably the version with no alignment restrictions on the squares).