An Engel Condition with Generalized Derivations on k-th Commutators

Let $R$ be a non-commutative ring of characteristic different from $2$, with center $Z(R)$, Utumi quotient ring $U$ and extended centroid $C$. Let $G$ be a non-zero generalized derivation of $R$, $k\geq 1$ a fixed integer, such that $[G([r_1,r_2]_k),[r_1,r_2]_k]=0$ for all $r_1, r_2 \in R$. Then one of the following holds: \begin{enumerate} \item there exists $\alpha \in C$ such that $G(x)=\alpha x$, for all $x\in R$; \item $R$ satisfies the standard identity $s_4(x_1,\ldots,x_4)$ and there exist $a\in U$, $\alpha \in C$ such that $G(x)=ax+xa+\alpha x$, for all $x\in R$.