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

On Kalai's conjectures concerning centrally symmetric polytopes