Schechtman's vector-balancing conjecture for zonotopes

Let $K\subseteq\mathbb{R}^d$ be a zonotope. For symmetric convex bodies, the vector balancing constant $\operatorname{vb}(K,K)$ is the least $r\geq0$ such that every finite sequence of vectors in $K$ admits signs whose signed sum lies in $rK$. Schechtman's conjecture.

$$\operatorname{vb}(K,K)\lesssim\sqrt d.$$

The source states this as Schechtman's question and notes that it would follow from the preceding conjectures. Its status is open.