Open, the site's label (OPEN). Three claim pages are recorded, two accepted partial claims and one accepted conditional claim, none of which settles the question. Harborth's five points and Kreisel and Kurz's seven points have the three properties, so the answer is yes for $n=5$ and for every $n\le7$, and no construction with eight points is known. Ascher, Braune and Turchet prove that Lang's conjecture, which is unproven, implies a uniform bound on the size of such sets, so that under it the answer would be no for all large $n$. Two further results settle no instance and are recorded here rather than as claims: Anning and Erdős [AnEr45] proved that an infinite set of points in the plane with all distances integers lies on a line, and Greenfeld, Iliopoulou and Peluse [GIP24] proved unconditionally, in their Corollary 1.3, that a set with the three properties inside $[-N,N]^2$ has $O((\log N)^{O(1)})$ points. The site's remarks also point to Problem 130.