Linear discrepancy conjecture for non-diagonal zero-sum-square-free matrices

Let $M=(a_{i,j})$ be an $n\times n$ matrix with entries in $\{-1,1\}$. Define its discrepancy by

$$\operatorname{disc}(M)=\sum_{1\leq i,j\leq n}a_{i,j}.$$

A matrix is zero-sum-square-free if it contains no square submatrix whose four entries sum to $0$, and it is diagonal if, after vertical and horizontal reflections, it becomes a $t$-diagonal matrix for some $t$.

Linear discrepancy conjecture. For every $C>0$, there is an integer $N$ such that, whenever $n\geq N$, every non-diagonal $n\times n$ matrix $M$ with entries in $\{-1,1\}$ and

$$|\operatorname{disc}(M)|\leq Cn$$

contains a zero-sum square.

Arévalo, Montejano and Roldán-Pensado proved this with the bound $n$ (apart from $n\leq4$), and remarked that their proof might yield $2n$. The conjecture asserts that every fixed linear bound $Cn$ eventually forces a zero-sum square in a non-diagonal matrix.