Komlós's conjecture for vector balancing

Let $A\in\mathbb{R}^{m\times n}$ be a matrix whose columns have Euclidean norm at most $1$, and let

$$\operatorname{disc}(A):=\min_{\sigma\in\{\pm1\}^n}\|A\sigma\|_\infty.$$

Komlós's conjecture. There exists a constant $K>0$ such that, for every such matrix $A$,

$$\operatorname{disc}(A)\leq K.$$

This is a major open problem in discrepancy theory. Banaszczyk's bound gives $\operatorname{disc}(A)=O(\sqrt{\log(\min\{m,n\})})$, and the conjecture contains the Beck–Fiala conjecture as a special case.