Towards a characterization of idempotent Schur multipliers

It is conjectured that every idempotent Schur multiplier can be written as a finite sum of contractive idempotents. This conjecture is equivalent to the statement that any boolean matrix $A$ with factorization norm $\lVert A\rVert_{\gamma_2}$ at most $\gamma$ can be expressed as a signed sum $$A = \sum_{i=1}^L \pm B_i,$$ where, up to permutation of rows and columns, each $B_i$ is a blow-up of an identity matrix, and $L$ depends only on $\gamma$. In this note we show that if $A$ is an $n\times n$ boolean matrix with $\lVert A\rVert_{\gamma_2} \le \gamma$, then it admits such an expression with $L = 2^{O(\gamma^9) + \log^*\! n}$, where $\log^*$ is the iterated logarithm function. As an application, any sequence of matrices with bounded factorization norm belongs to the complexity class $\mathrm{P}^\mathrm{EQ}$ of communication problems with polylogarithmic equality-oracle complexity.