  We prove that in the first Heisenberg group, unlike Euclidean spaces and
higher dimensional Heisenberg groups, the best possible exponent for the strong
geometric lemma for intrinsic Lipschitz graphs is $4$ instead of $2$. Combined
with earlier work from arXiv:2004.11447 and arXiv:2207.03013, our result
completes the proof of the strong geometric lemma in Heisenberg groups. One key
tool in our proof, and possibly of independent interest, is a suitable
refinement of the foliated coronizations which first appeared in
arXiv:2004.12522.
