On the cone of \(f\)-vectors of cubical polytopes

What is the minimal closed cone containing all \(f\)-vectors of cubical \(d\)-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical \(g\)-vector coordinates, contains the nonnegative \(g\)-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.