On solvable groups of exponent 4

Given an arbitrary identity $v=1$, there exists a positive integer $N=N(v)$ such that for every metabelian group $G$ and every generating set $A$ for $G$ the following holds: If each subgroup of $G$ generated by at most $N$ elements of $A$ satisfies the identity $v=1$ then the group $G$ itself satisfies this identity. A similar assertion fails for center-by-metabelian groups. This answers Bludov's question.