KöMaL 2016 B4838

A centrally symmetrical convex polyhedron $\mathcal{P}$ has the following property: any two vertices $X$ and $Y$ of $\mathcal{P}$ are either centrally opposite or there exists a face of $\mathcal{P}$ that contains both. Show that $\mathcal{P}$ is either a parallelepiped, or the vertices of $\mathcal{P}$ are the centres of the faces of a parallelepiped.