A survey by Nguyen Van Thé (2014) 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?