  Almost finiteness was introduced in the seminal work of Kerr as an dynamical
analogue of Z-stability in the Toms-Winter conjecture. In this article, we
provide the first examples of minimal, topologically free actions of amenable
groups that have mean dimension zero but are not almost finite. More precisely,
we prove that there exists an infinite family of amenable wreath products that
admit topologically free, minimal profinite actions on the Cantor space which
fail to be almost finite. Furthermore, these actions have dynamical comparison.
This intriguing new phenomenon shows that Kerr's dynamical analogue of
Toms-Winter conjecture fails for minimal, topologically free actions of
amenable groups.
  The notion of allosteric group holds a significant position in our study. A
group is allosteric if it admits a minimal action on a compact space with an
invariant ergodic probability measure that is topologically free but not
essentially free. We study allostery of wreath products and provide the first
examples of allosteric amenable groups.
