Centralizers of generalized skew derivations on multilinear polynomials

Let $\mathscr R$ be a prime ring of characteristic different from $2$, let $\mathscr Q$ be the right Martindale quotient ring of $\mathscr R$, and let $\mathscr C$ be the extended centroid of $\mathscr R$. Suppose that $\mathscr G$ is a nonzero generalized skew derivation of $\mathscr R$ and $f(x_1,\dots,x_n)$ is a noncentral multilinear polynomial over $\mathscr C$ with $n$ noncommuting variables. Let $f(\mathscr R)=\{f(r_1,\dots,r_n)\colon r_i\in\mathscr R\}$ be the set of all evaluations of $f(x_1,\dots,x_n)$ in $\mathscr R$, while $\mathscr A=\{[\mathscr G(f(r_1,\dots,r_n)),f(r_1,\dots,r_n)]\colon r_i\in\mathscr R\}$, and let $C_\mathscr R(\mathscr A)$ be the centralizer of $\mathscr A$ in $\mathscr R$; i.e., $C_\mathscr R(\mathscr A)=\{a\in\mathscr R\colon[a,x]=0\ \forall x\in\mathscr A\}$. We prove that if $\mathscr A\neq(0)$, then $C_\mathscr R(\mathscr A)=Z(R)$.