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Strongly maximal chains in FAC posets

Prove that every poset P with no infinite antichain (i.e., every FAC poset) contains a strongly maximal chain C such that for every chain D in P, the inequality |C \ D| ≥ |D \ C| holds.

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Background

The paper studies structural properties of posets with no infinite antichains (FAC posets) and introduces strongly maximal chains (SMCs) as duals to strongly maximal antichains established under related conditions. The authors prove that every countable FAC poset has an SMC, but they cannot resolve the general case, leading to this conjecture.

An SMC is a maximal chain C such that for any other chain D, C is at least as large outside D as D is outside C. Establishing existence of SMCs in all FAC posets would generalize their countable result and parallel known results for strongly maximal antichains.

References

It is natural to consider the dual of \Cref{thm:strongly-maximal-antichains}, and thus arrive at the following conjecture. If $P$ is an FAC poset, then $P$ contains a strongly maximal chain.

— A resolution of the Aharoni-Korman conjecture (2411.16844 - Hollom, 25 Nov 2024) in Conjecture 1.3, Section 1.3 (Our contributions)