  In 1989 Kalai stated the three conjectures A, B, C of increasing strength
concerning face numbers of centrally symmetric convex polytopes. The weakest
conjecture, A, became known as the ``$3^d$-conjecture''. It is well-known that
the three conjectures hold in dimensions d \leq 3. We show that in dimension 4
only conjectures A and B are valid, while conjecture C fails. Furthermore, we
show that both conjectures B and C fail in all dimensions d \geq 5.
