Antichains in partially ordered sets of singular cofinality

In their paper from 1981, Milner and Sauer conjectured that for any poset < P, <=>, if cf (P, <=) = lambda > cf(lambda) = kappa, then P must contain an antichain of size kappa. We prove that for lambda > cf(lambda) = kappa, if there exists a cardinal mu < lambda such that cov(lambda, mu, kappa, 2) = lambda, then any poset of cofinality lambda contains lambda(kappa) antichains of size.. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.