This is crossposted from MSE.

We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$.

In his article Monotileable Amenable Groups, B. Weiss gives lots of examples of amenable groups which admit a left-Følner sequence of monotiles (e.g. profinite or solvable groups). Is there an example of a countable amenable group which is known not to have a left-Følner sequence of monotiles?