Subcontinuum bridging property between two points in ultrafilter order

Determine whether, for every chainable continuum X, every ultrafilter order ≤_U^D on X, and any two distinct points a,b ∈ X, there exists a non-degenerate subcontinuum K ⊆ X such that for all x ∈ K one has a ≤_U^D x ≤_U^D b.

Background

This question strengthens the density inquiry by asking not only for a point between a and b, but for a whole non-degenerate subcontinuum whose points lie between a and b in the ultrafilter order.

Such a property would connect order-theoretic density with geometric structure in chainable continua, potentially revealing deeper interactions between the ultrafilter order and the continuum’s subcontinua.