Linearly ordered coarse spaces

A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either $0$ or $1$. If $X$ is metrizable then the family of all right bounded subsets of $X$ has a selector.