The bounded blocky-matrix complexity conjecture for bounded max-norm

Let $M$ be a Boolean matrix, meaning a matrix with entries in $\{0,1\}$. Its factorization max-norm is denoted by $\gamma_2(M)$, and let $\operatorname{bl}(M)$ be the minimum number of blocky matrices whose $\pm1$-linear combination is $M$. The bounded blocky-matrix complexity conjecture. For every $\gamma>0$ there exists $b_{\gamma}$ such that every Boolean matrix $M$ with $\gamma_2(M)\leq\gamma$ satisfies $\operatorname{bl}(M)\leq b_{\gamma}$. This conjecture asks for a qualitative converse to the inequality $\gamma_2(M)\leq\operatorname{bl}(M)$ and is equivalent to Conjecture III of HHH. Its resolution status is not specified in the source.