Does small γ₂-norm imply a large monochromatic rectangle?

Determine whether there exists a function κ(c) > 0 such that for every Boolean matrix F with γ₂(F) ≤ c, the rectangle ratio satisfies rect(F) ≥ κ(c).

Background

Cohen-type structure theorems for group-lifts imply that bounded γ₂-norm enforces a signed sum of few blocky matrices, which in turn would force large monochromatic rectangles. For general Boolean matrices (beyond group-lifts), it is unclear whether bounded γ₂-norm alone yields any quantitative rectangle guarantee.

An affirmative answer would follow from the blocky decomposition conjecture and would have consequences for deterministic and oracle-based communication models.