A metrizable completely regular ordered space

summary:We construct a completely regular ordered space $(X,{\Cal T},\leq)$ such that $X$ is an $I$-space, the topology $\Cal T$ of $X$ is metrizable and the bitopological space $(X,{\Cal T}^\sharp,{\Cal T}^{\flat})$ is pairwise regular, but not pairwise completely regular. (Here ${\Cal T}^\sharp$ denotes the upper topology and ${\Cal T}^\flat$ the lower topology of $X$.