Words on finite nilpotent groups of class 2

Let $G$ be a finite nilpotent group of class at most $2$, and let $w=w(x_1,\ldots,x_n)$ be a group word in $n$ variables. Then we prove that the number of solutions in $G\times\overset{n}{\cdots}\times G$ to the equation $w=1$ is at least $|G|^{n-1}$. This result, also independently obtained bt Matthew Levy [L], solves a special case of a conjecture of Alon Amit.