External cofinalities and the antichain condition in partial orders

Does every partial order of singular cofinality λ have an antichain of size ? This is the Singular Cofinality Conjecture. M. Pouzet proved [M. Pouzet, Parties cofinales des ordres partiels ne contenant pas d’antichaines infinies, 1980, preprint] that there must be an infinite antichain. When is uncountable, the positive answer is only consistently true, but unknown in ZFC. In this note we investigate this question from the purely set-theoretic point of view. On the way, we answer a question of Milner and Pouzet from [E.C. Milner, M. Pouzet, Posets with singular cofinality, 1997, preprint].