Bounded max-norm matrices and blocky decompositions

Prove that for every b[?]positive real number b[?]gamma there exists a constant b_gamma such that every Boolean matrix M with gamma_2(M) at most gamma can be expressed as a plus-or-minus one linear combination of at most b_gamma blocky matrices.

Background

A blocky matrix is a Boolean matrix whose one entries form disjoint complete rectangular blocks after suitable row and column partitions. The quantity block(M) is the minimum number of blocky matrices required in a plus-or-minus one linear combination representing an integer matrix M.

The paper establishes partial progress toward this conjecture for locally sparse matrices: if a bounded-gamma_2 Boolean matrix avoids a fixed-size all-ones submatrix, then it has bounded average degree and, consequently, admits a bounded decomposition into thin blocky matrices. The unresolved problem is whether the same qualitative bounded decomposition holds for all Boolean matrices of bounded gamma_2-norm.