Let v1, . . . , vn be vectors in Rm of Euclidean norm at most one. The Koml´os signing prob-lem asks for signs ϵj ∈ {−1, 1} such that the signed sum has small ℓ∞ norm, with a boundindependent of m and n. A recent result of Guo, Fang, and Lu establishes such a bound withconstant 3√2π ≈ 7.5199, via a directional total variation invariant and a stability lemma whosethreshold is κ∥v∥2 ≤ 1/3. The threshold is a consequence of a height identity that controls themass retained by a symmetrized lift of the density under the Banaszczyk transform, togetherwith a linear estimate on the translation distance of the density. We replace the linear estimatewith a sharper bound derived from the exact Hellinger affinity of the product cosine-squareddensity, combined with the linear bound. This gives an improved constant C′ ≈ 7.4858. Theresidual gap to the asymptotic value C⋆ ≈ 6.9013 of the method is characterized by a Gaus-sian comparison conjecture for localized product experiments; we prove the case k = 1 of thatconjecture in the range used by the reduction.