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Density of ultrafilter orders defined via chains (conjecture)

Prove that for every chainable continuum X, every ultrafilter order ≤_U^D defined via a sequence of chains D covering X (with mesh(D_n) → 0) and a non-principal ultrafilter U on ℕ is a dense linear order; that is, for all a,b ∈ X with a <_U^D b there exists c ∈ X such that a <_U^D c <_U^D b.

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Background

The authors show that ultrafilter orders arising from inverse-limit representations are dense, and they conjecture that the chain-based ultrafilter orders share this property.

Establishing density would strengthen the structural understanding of these orders and align the two construction methods (chains and inverse limits) in an important qualitative aspect.

References

One can easily show that ultrafilter orders on the inverse limit homeomorphic to some chainable continuum (in the sense of Definition \ref{inv_lim}) are dense. We have a conjecture that ultrafilter orders defined using sequence of chains obtained from chainability of X are also dense orders.

— Linear orders on chainable continua (2510.14577 - Marciszewski et al., 16 Oct 2025) in Section 7 (Questions)