Triplets of pure free squarefree complexes

On the category of bounded complexes of finitely generated free squarefree modules over the polynomial ring S, there is the standard duality functor D = Hom_S(-, omega_S) and the Alexander duality functor A. The composition AD is an endofunctor on this category, of order three up to translation. We consider complexes F of free squarefree modules such that both F, AD(F) and (AD)^2(F) are pure, when considered as singly graded complexes. We conjecture i) the existence of such triplets of complexes for given triplets of degree sequences, and ii) the uniqueness of their Betti numbers, up to scalar multiple. We show that this uniqueness follows from the existence, and we construct such triplets if two of them are linear.