Non-locality of the meet levels of the Trotter-Weil Hierarchy

We prove that the meet level $m$ of the Trotter-Weil, ${\mathsf{V}}_m$ is not local for all $m\ge 1$, as conjectured in a paper by Kufleitner and Lauser. In order to show this, we explicitly provide a language whose syntactic semigroup is in $L{\mathsf{V}}_m$ and not in ${\mathsf{V}}_m\ast \mathsf{D}$.