Schechtman's zonotope sparsification conjecture

Let $Z\subseteq\mathbb R^d$ be a $d$-dimensional zonotope and let $\varepsilon\in(0,1)$. A zonotope is represented as a Minkowski sum of line segments; in particular, write

$$\widetilde Z=\widetilde A^\top[-1,1]^{\widetilde m}\subseteq\mathbb R^d.$$

Schechtman's sparsification conjecture. Is there a universal constant $C$ such that there is a zonotope $\widetilde Z$ generated by at most

$$\widetilde m\le \frac{Cd}{\varepsilon^2}$$

segments and satisfying

$$\widetilde Z\subseteq Z\subseteq(1+\varepsilon)\widetilde Z?$$

This is presented as a version of Schechtman's 1987 sparsification problem. The source gives no resolution, so the problem remains open.