Multilinear polynomials and cocentralizing conditions in prime rings

Let $R$ be a noncommutative prime ring of characteristic different from 2, let $Z(R)$ be its center, let $U$ be the Utumi quotient ring of $R$, let $C$ be the extended centroid of $R$, and let $f(x_1,\dots,x_n)$ be a noncentral multilinear polynomial over $C$ in $n$ noncommuting variables. Denote by $f(R)$ the set of all evaluations of $f(x_1,\dots,x_n)$ on $R$. If $F$ and $G$ are generalized derivations of $R$ such that $[[F(x),x],[G(y),y]]\in Z(R)$ for any $x,y\in f(R)$, then one of the following holds:
(1) there exists $\alpha\in C$ such that $F(x)=\alpha x$ for all $x\in R$;
(2) there exists $\beta\in C$ such that $G(x)=\beta x$ for all $x\in R$;
(3) $f(x_1,\dots,x_n)^2$ is central valued on $R$ and either there exist $a\in U$ and $\alpha\in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$ or there exist $c\in U$ anf $\beta\in C$ such that $G(x)=cx+xc+\beta x$ for all $x\in R$;
(4) $R$ satisfies the standard identity $s_4(x_1,\dots,x_4)$ and either there exist $a\in U$ and $\alpha\in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$ or there exist $c\in U$ and $\beta\in C$ such that $G(x)=cx+xc+\beta x$ for all $x\in R$.