Online Banaszczyk conjecture for online vector balancing

Let $v_1,\ldots,v_T\in\mathbb{R}^m$ be a sequence of vectors with Euclidean norm at most $1$. In the oblivious adversarial online setting, the algorithm receives $v_t$ at time $t$ and outputs a sign $\sigma_t\in\{\pm1\}$. Online Banaszczyk conjecture. There exists a randomized algorithm that, with high probability, achieves

$$\max_{t\in[T]}\left\lVert\sum_{s=1}^t\sigma_s v_s\right\rVert_\infty\lesssim\sqrt{\log(mT)}.$$

This conjecture asks for an online analogue of Banaszczyk's discrepancy bound. The source attributes it to Alweiss, and presents it as an open conjecture concerning efficient online balancing against an oblivious adversary.