Belt distance between facets of space-filling zonotopes

For every d-dimensional polytope P with centrally symmetric facets we can associate a "subway map" such that every line of this "subway" corresponds to set of facets parallel to one of ridges P. The belt diameter of P is the maximal number of line changes that you need to do in order to get from one station to another. In this paper we prove that belt diameter of d-dimensional space-filling zonotope is not greater than $\lceil \log_2\frac45d\rceil$. Moreover we show that this bound can not be improved in dimensions d at most 6.