gr.group theory - А generalization of Gromov's theorem on polynomial growth

My paper with Shalom does settle the question when $S = S_n$ is known to have size polynomial in n (and maybe is allowed to grow just a little bit faster than this, something like $n^{(log log n)^c}$ or so), but I doubt that the result is known yet if S is allowed to be arbitrarily large.

Note that even the bounded case is nontrivial - it's not obvious why having $|S_n^n| leq n^{O(1)} |S_n|$ implies polynomial growth. (There is no reason why growth has to be uniform for fixed cardinality of generators; for instance, I believe it is a major open problem (due to Gromov?) as to whether exponential growth is the same as uniform exponential growth for finitely generated groups.)

If we had a good non-commutative Freiman theorem, then one may possibly be able to settle your question affirmatively (note from the pigeonhole principle that if $|S_n^n| leq n^C |S_n|$ for some large n, then there exists an intermediate $m=m_n$ between 1 and n such that the set $B := S_n^m$ has small doubling, in the sense that $|B^2| = O(|B|)$). The best result in this direction for general groups currently is due to Hrushovski, which does show that sets of small doubling contain some vaguely "virtually nilpotent" structure, but it is not yet enough to give Gromov's theorem, let alone the generalisation you mention above. More is known if one already has some additional structure on the group (e.g. it has a faithful linear representation of bounded dimension, or if it is already known to be virtually solvable).