Non-separability of the Lipschitz distance

Let $X$ be a compact metric space and ${\mathcal{M}}_X$ be the set of isometry classes of compact metric spaces $Y$ such that the Lipschitz distance $d_L(X,Y)$ is finite. We show that $({\mathcal{M}}_X,d_L)$ is not separable when $X$ is a closed interval, or an infinite union of shrinking closed intervals.