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SMCs in FAC posets of size aleph_n

Prove that for each natural number n, every poset P with no infinite antichain and |P| = aleph_n contains a strongly maximal chain.

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Background

The authors establish existence of strongly maximal chains in countable FAC posets and conjecture extensions to larger regular cardinals of the form aleph_n. They believe the technical obstacles for these cardinals are surmountable, potentially via induction on n.

Confirming this conjecture would provide a broad structural principle for SMC existence across all finite stages of the aleph hierarchy.

References

While there appear to be further technical difficulties in extending the work to non-scattered posets of cardinality at least $\aleph_\omega$, the author does not believe these to be insurmountable, and that the following conjecture should be amenable to an induction on the cardinality. Let $P$ be an FAC poset of cardinality $\aleph_n$ for some $n<\omega$. Then $P$ has a strongly maximal chain.

— A resolution of the Aharoni-Korman conjecture (2411.16844 - Hollom, 25 Nov 2024) in Conjecture 7.3, Section 7 (Concluding remarks and open problems)