Strictly complete regularity of the Lawson topology on a continuous poset

For a continuous poset P, let ω(P) and λ(P) be the lower topology and the Lawson topology on P, respectively. In this paper, we constructively prove in the set theory ZFDCω (the theory obtained by adjoining the axiom of ω dependent choices to the Zermelo–Fraenkel set theory) that if all lower closed subsets in (P,λ(P)) are closed in (P,ω(P)), then (P,λ(P)) is a strictly completely regular ordered space.