On the de Morgan property of the standard Brouwer-Zadeh poset

The standard Brouwer-Zadeh poset Sigma (H) is the poset of all effect opera tors on a Hilbert space H. naturally equipped with two types of orthocomple mentation .In developing the theory , the question occured if (when) Sigma (H) fulfils the de Morgan property with respect to both orthocomplementatio n operations. In Ref.3 the authors proved that it is the case provided dim H<<proportional to>, and they conjectured that if dim H=proportional to, th en the answer is in the negative. In this note, we first give a somewhat si mple proof of the known result for dim H<<proportional to>, and then we giv e a proof to the conjecture. We show that if dim H=proportional to, then th e de Morgan property is not valid.