[A survey by Nguyen Van Thé (2014)](https://arxiv.org/abs/1412.3254v2) has Conjecture 1,
which is that
"every closed oligomorphic
subgroup of $S\_∞$ should have a metrizable universal minimal flow with a generic
orbit." Later, it goes on to say that "it is even possible that this should be
true for a larger class of groups, called Roelcke precompact." (Let me call this Conjecture 1'.) Now, Kwiatkowska (2018) exhibited a group without a metrizable universal minimal flow that is not Roelcke precompact, so we need to stick to Conjecture 1.


How about the converse of Conjecture 1', i.e., the statement that a closed subgroup of $S\_∞$ is Roelcke precompact if it has a metrizable universal minimal flow? Is there a proof or a counterexample? In absence of either of the two, do people believe it?



Roelcke precompactness and Ramsey property
The isometry group of the ordered rational Urysohn space is extremely amenable but not Roelcke precompact, which gives a counterexample.


