Centrally symmetric neighborly polytopes

Let $$P$$ be a centrally symmetric $$k$$-neighborly $$d$$-polytope. Let $$P^*$$ is polar (also dual) of $$P$$. Consider a polytope $$Q^*$$ , $$Q^* \subset P^*$$(not necessarily a face of $$P^*$$). By duality, $$P\subset Q$$. How neighbourly is this $$Q$$? Is it $$k$$, or less than $$k$$, or more than $$k$$-neighborly?

• This does not answer your question, but Grünbaum's book Convex Polytopes indicates (p.129b) that it is an open problem whether there exists any polytope other than a simplex that is 2-neighborly and its dual is 2-neighborly. – Joseph O'Rourke Mar 28 '12 at 19:34
• The question is not clear. What do you assume on Q? is it a convex hull of some vertices of P? – Gil Kalai Dec 30 '13 at 9:37