Non-connectedness and non-compactness of ultrafilter order topologies on indecomposable chainable continua

Establish whether, for every indecomposable chainable continuum X and every ultrafilter order ≤_U^D on X, the order topology τ_U^D is non-connected and non-compact.

Background

The authors show that for the Knaster continuum, under a certain ultrafilter order, the induced order topology is non-connected and non-compact. They ask whether these properties persist for all indecomposable chainable continua under arbitrary ultrafilter orders.

A general affirmative result would strengthen the topological characterization of ultrafilter order topologies across a broad class of indecomposable chainable continua.