Almost-cover conjecture for zonotopes

Let $Z$ be a zonotope, and let $\operatorname{rk}(Z)$ denote its rank, namely the number of summands in a representation of $Z$ as the Minkowski sum of a nondegenerate collection of line segments. An almost cover of the vertices of $Z$ is a collection of hyperplanes whose union contains all vertices except one and does not contain that remaining vertex. Zonotope almost-cover conjecture. Every almost cover of the vertices of a zonotope $Z$ consists of at least $\operatorname{rk}(Z)$ hyperplanes. The source gives supporting examples, including centrally symmetric polygons and the cube, but does not report a proof or disproof of the general assertion.