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,≤〉=λ>cfλ=κ, then P must contain an antichain of size κ. We prove that for λ > cf(λ)=κ, if there exists a cardinalμ<λ such that cov(λ, μ, κ, 2)=λ, then any poset of cofinality λ contains λ ^κ antichains of size κ. 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.