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SMCs in scattered FAC posets

Prove that every scattered poset P with no infinite antichain contains a strongly maximal chain.

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Background

Scattered posets exclude dense suborders like the rationals, and the paper shows several positive results for scattered FAC posets (e.g., existence of SMCs in the countable case). Motivated by these, the authors conjecture that scattering alone, together with the FAC condition, suffices for SMC existence at any cardinality.

Establishing this would significantly generalize the countable case and align strongly maximal chain existence with widely used structural constraints in order theory.

References

In the scattered case however, there seems to be more hope, and so we also make the following conjecture. Let $P$ be a scattered FAC poset. Then $P$ has a strongly maximal chain.

— A resolution of the Aharoni-Korman conjecture (2411.16844 - Hollom, 25 Nov 2024) in Conjecture 7.4, Section 7 (Concluding remarks and open problems)