Suppose that we have a group $G$ generated by a set $S$ of elements with the only family of relations being that all commutators are central. Equivalently, $G$ is the largest group of nilpotency class $2$ generated by $S$. Now suppose that we have two equivalent words $x_1x_2\cdots x_n$ and $y_1y_2\cdots y_n$, where each $x_i$ and $y_j$ is a generator. Does it follow that $x_i = y_i$ for all $i$?

Intuitively, the answer should be "no", since that's the usual answer to these kinds of questions, but I am having trouble constructing a counterexample.