Strong Komlós conjecture for prefix discrepancy in the Euclidean ball

Let $T$ be a positive integer, let $v_1,\ldots,v_T\in\mathbb{B}_2^d$ be adversarial input vectors, and let $[T]=\{1,\ldots,T\}$. A signing is a vector $x\in\{\pm1\}^T$. Strong Komlós conjecture. There always exists a signing $x\in\{\pm1\}^T$ such that

$$\max_{\tau\in[T]}\left\|\sum_{t\leq\tau}x_t v_t\right\|_2=O(\sqrt d).$$

This asks for a dimension-dependent prefix-discrepancy bound for adversarial vectors in the Euclidean unit ball. The paper notes that related prefix-discrepancy questions have been posed for other norms; the status evidence identifies the Beck–Fiala version as another related open problem, and this conjecture remains open.