On a conjecture on the Sperner property (Q1264161)

A finite partially ordered set P has the Sperner property if the maximum cardinality of antichains in P equals the maximum cardinality of sets of elements in P with the same rank. \textit{Ko-Wei Lih} [J. Comb. Theory, Ser. A 29, 182-185 (1980; Zbl 0446.05002)] proved that if P is the set of all subsets of an n-element set with set inclusion as order relation, and F is an order filter in P generated by a collection of elements in P with the same rank \(t=1\), then F has the Sperner property; he conjectured also that the same holds for \(t>1\). Here it is shown by counterexamples that this conjecture is false if \(t\geq 4\) and that it is right if \(t=2\) or \(t=3\) in some special cases.