Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows. Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with pointwise multiplication to make it a group. The wreath product $A\wr B$ is the semidirect product $A^{(B)}\ltimes B$. Therefore the elements of $A\wr B$ are pairs $(f,b)$ where $f\in A^{(B)}$ and $b\in B$. Multiplication in $A\wr B$ is given by $$(f,b)(g,c)=(fg^b,bc)$$ where $g^b(x)=g(b^{-1}x)$ for every $x\in B$.
Now let $A$ and $B$ be finitely generated free abelian group and $W=A\wr B$. Is this true that every non-cyclic abelian subgroup is contained in $A^{(B)}$ or a conjugate of $B$?