Milner–Sauer conjecture on antichains in posets of singular cofinality

Let $\langle P,\le\rangle$ be a poset. Write $\operatorname{cf}(P,\le)$ for its cofinality, and let $\lambda$ and $\kappa$ be cardinals satisfying

$$\operatorname{cf}(P,\le)=\lambda>\operatorname{cf}(\lambda)=\kappa.$$

An antichain in $P$ is a subset whose distinct elements are pairwise incomparable.

Milner–Sauer conjecture. If $\operatorname{cf}(P,\le)=\lambda>\operatorname{cf}(\lambda)=\kappa$, then $P$ contains an antichain of size $\kappa$.

The conjecture asks how large an antichain must occur in a poset whose cofinality is singular. The paper proves the conclusion under the additional hypothesis that there is a cardinal $\mu<\lambda$ with $\operatorname{cov}(\lambda,\mu,\kappa,2)=\lambda$; the consistency of the negation of that hypothesis is stated to be unknown.