On automorphisms of the relatively free groups satisfying the identity $[x^n,~y] = 1$

We prove that if an automorphism $\varphi$ of the relatively free group of the group variety, defined by the identity relation $[x^n,~y] = 1$, acts identically on its center, then $\varphi$ has either infinite or odd order, where $n\geq665$ is an arbitrary odd number.