Question

Let $P$ is 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?