The fair-discrepancy propagation conjecture for fully balanced matrices

Let $A\in\mathbb{M}_{n\times m}(\mathbb{R})$ be a fully balanced matrix. For each row $i$, let

$$M_i=\frac{1}{m}\sum_{j=1}^{m}a_{ij}$$

be its average discrepancy. If, for a fixed $1\leq i\leq n$, one has

$$|M_i-a_{ij}|<\epsilon$$

for the relevant entries $j$, then $|M_i-a_{ij}|<\epsilon$ for all $1\leq i\leq n$.

Fair-discrepancy propagation conjecture. If one fixed row has discrepancy less than $\epsilon$, then every row has discrepancy less than $\epsilon$.

The claim concerns propagation of a fair discrepancy from one row to all rows of a fully balanced matrix. The supplied text gives no resolution or additional context.