Let $F$ and $G$ be two real-valued probability distributions, considered as cadlag functions with the uniform norm [not the Skorohod metric], and let $\mathbb{F}_n$ and $\mathbb{G}_m$ be empirical distributions of samples of size $n$ from $F$ and $m$ from $G$. If $\preceq$ is a continuous total preorder, the Glivenko-Cantelli theorem shows that $F\prec G$ implies $P(\mathbb{F}_n\prec\mathbb{G}_m)\to 1$. That is, continuity of the preorder implies consistency of the test
Is the converse true? That is, from consistency of the test, $F\prec G$ implies $P(\mathbb{F}_n\prec\mathbb{G}_m)\to 1$, can we deduce continuity of the preorder? This wouldn't be true for the Skorohod metric, but I haven't been able to find a counterexample with the uniform norm.