Completely regularly ordered spaces versus T2-ordered spaces which are completely regular

Schwarz and Weck-Schwarz have shown that a T2-ordered space (X,τ,⩽) whose underlying topological space (X,τ) is completely regular need not be a completely regularly ordered space (that is, T3.5+T2-ordered ⇏T3.5-ordered). We show that a completely regular T2-ordered space need not be completely regularly ordered even under more stringent assumptions such as convexity of the topology. One example involves the construction of a nontrivial topological ordered space on which every continuous increasing function into the real unit interval is constant.