infer the expansion of the Universe without distances

As I note in this blog post, it is possible to infer that the Universe is expanding, even if you have only velocities and no distances. The idea is that you would marginalize out all distances and the Hubble Constant, and do a likelihood test (or equivalent, like, say, cross-validation). The two hypotheses would be a gas of galaxies with finite velocity dispersion but a well-defined mean rest-frame velocity through which we are moving (at unknown speed) vs an expanding gas, again with the finite velocity dispersion and through which we are moving. I think the test is very easy to set up and the problem is very easy to solve, and would demonstrate that it only takes a handful of velocities (but very good sky coverage!) to demonstrate expansion.