  Given a matrix $M = (a_{i,j})$ a square is a $2 \times 2$ submatrix with
entries $a_{i,j}$, $a_{i, j+s}$, $a_{i+s, j}$, $a_{i+s, j +s}$ for some $s \geq
1$, and a zero-sum square is a square where the entries sum to $0$. Recently,
Ar\'evalo, Montejano and Rold\'an-Pensado proved that all large $n \times n$
$\{-1,1\}$-matrices $M$ with discrepancy $|\sum a_{i,j}| \leq n$ contain a
zero-sum square unless they are split. We improve this bound by showing that
all large $n \times n$ $\{-1,1\}$-matrices $M$ with discrepancy at most $n^2/4$
are either split or contain a zero-sum square. Since zero-sum square free
matrices with discrepancy at most $n^2/2$ are already known, this bound is
asymptotically optimal.
