The bounded row and column sum discrepancy conjecture

Let $V$ be an $n \times m$ real matrix with $|V_{i,j}| \leq 1$, $\|v^i\|_1 \leq R$, and $\|v_j\|_1 \leq \Delta$ for all $i \in [n]$ and $j \in [m]$. Assume $R \geq \Delta$. The bounded row and column sum discrepancy conjecture. There exists $y \in \{-1,+1\}^m$ such that

$$\|Vy\|_\infty \leq O(\sqrt{R}).$$

This conjecture asks whether the logarithmic factor in the paper’s discrepancy bounds can be removed. The source does not provide evidence that the conjecture has been resolved.