Openness of composants in the ultrafilter order topology on indecomposable chainable continua

Ascertain whether, for every chainable indecomposable continuum X and every ultrafilter order ≤_U^D on X, each composant of X is open in the order topology τ_U^D generated by ≤_U^D.

Background

In the Knaster continuum example, composants coincide with arc components and are open in the order topology generated by certain ultrafilter orders. The authors ask whether this openness extends to all indecomposable chainable continua under arbitrary ultrafilter orders.

A positive answer would provide a general structural insight into how ultrafilter orders interact with the decomposition of indecomposable chainable continua into composants.