Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the same)?

Does there exist a smooth proper complex variety whose analytification is not homotopy equivalent to the analytification of a smooth projective complex variety?