Bounded $cgamma_2$-norm decomposition conjecture for Boolean matrices

Let $M$ be an $m\times n$ Boolean matrix, and let $\gamma_2(M)$ denote its $\gamma_2$-norm. A blocky matrix is a blow-up of a permutation matrix. Bounded $\gamma_2$-norm decomposition conjecture. If

$$\gamma_2(M)\leq \gamma,$$

then there exists a constant $c_{\gamma}$ depending only on $\gamma$ such that $M$ is a $\pm1$-linear combination of at most $c_{\gamma}$ blocky matrices.

The conjecture is equivalent, according to the source, to a structural characterization of matrices with bounded $\gamma_2$-norm and to an open question about characterizing idempotent Schur multipliers in terms of contractive idempotents.