Zero-Sum Squares in Bounded Discrepancy $\{-1,1\}$-Matrices

For $n\geqslant 5$, we prove that every $n\times n$ matrix $\mathcal{M}=(a_{i,j})$ with entries in $\{-1,1\}$ and absolute discrepancy $\lvert\mathrm{disc}(\mathcal{M})\rvert=\lvert\sum a_{i,j}\rvert\leqslant contains a zero-sum square except for the split matrices (up to symmetries). Here, is $2\times 2$ sub-matrix of $\mathcal{M}$ $a_{i,j}, a_{i+s,s}, a_{i,j+s}, a_{i+s,j+s}$ some $s\geqslant 1$, all above diagonal equal $-1$ remaining $1$. In particular, show 5$ square.