The large constant submatrix conjecture for bounded max-norm matrices

Let $M$ be an $m\times n$ binary matrix, with entries in $\{0,1\}$, and let $\gamma_2(M)$ denote its factorization max-norm. A submatrix is obtained by restricting $M$ to selected sets of rows and columns. The large constant submatrix conjecture. For every $C>0$ there exists $c>0$ such that the following holds: if $\gamma_2(M)\leq C$, then $M$ contains a $cm\times cn$ submatrix that is either all ones or all zeros. The source presents this as an easier consequence to pursue independently of the bounded blocky-matrix conjecture; its resolution status is not specified.