Centralizers of generalized derivations on multilinear polynomials in prime rings

Let $R$ be a prime ring of characteristic different from 2, with Utumi quotient ring $U$ and extended centroid $C$, $\delta$ a nonzero derivation of $R$, $G$ a nonzero generalized derivation of $R$, and $f(x_1,\dots,x_n)$ a noncentral multilinear polynomial over $C$. If $\delta(G(f(r_1,\dots,r_n))f(r_1,\dots,r_n))=0$ for all $r_1,\dots,r_n\in R$, then $f(x_1,\dots,x_n)^2$ is central-valued on $R$. Moreover there exists $a\in U$ such that $G(x)=ax$ for all $x\in R$ and $\delta$ is an inner derivation of $R$ such that $\delta(a)=0$.