The lamplighter exclusion conjecture for diagram groups

Let $G$ be a diagram group, and let $\mathbb{Z} \wr \mathbb{Z}^2$ denote the wreath product of $\mathbb{Z}$ by $\mathbb{Z}^2$. Lamplighter exclusion conjecture. The group $G$ does not contain $\mathbb{Z} \wr \mathbb{Z}^2$. The claim is posed in the discussion of solvable diagram groups, where only a few examples are known; determining which metabelian or solvable groups occur as diagram groups remains open.