Let $G$ be a finitely generated torsion-free nilpotent group. Let $x\in G$.
Assume that $[a,[a,x]]=1$ for every $a\in G$.
Does it follow that $[a,x]\in Z(G)$ for every $a\in G$?
My feeling is that the answer should be "no". My intuition is that if you take $K$ to be a finitely generated free nilpotent group of class $3$, take $x$ to be one of the generators of $K$, and let $G$ be the quotient of $K$ by the normal closure of the set of elements of the form $[a,[a,x]]$ (running over $a\in G$), then I see no reason this normal closure would contain all elements $[b,[a,x]]$ (running over $a,b\in G$). But this is, of course, not a proof.
NOTE: If there is a counterexample $G$, I would especially like to have a counterexample of class $3$.