**Komlos Conjecture**: the exists an absolute constant $K>0$ such that for all $d$ and any collection of vectors $v_1,\ldots, v_n\in \mathbb{R}^d$ with $\left\lVert v_i\right\rVert _2=1$ we can find weights $w_i\in\{-1,1\}$ such that $\left\lVert w_1v_1+\cdots+w_nv_n\right\rVert _{\infty}<K$.

I would like to ask what the current state of this conjecture is? Is it true that the best result towards this conjecture is that for fixed $d$ one can show the latter statement with $K=K(n)\approx \log n$?