On $m$-commuting mappings with skew derivations in prime rings

Let $m,k$ be two fixed positive integers, $R$ a prime ring with the Martindale qoutient ring $Q$, $L$ a noncommutative Lie ideal of $R$, and $\delta$ a skew derivation of $R$ associated with an automorphism $\varphi$, denoted by $(\delta,\varphi)$. If $[\delta(x),x^m]_k=0$ for all $x\in L$, then $char(R)=2$ and $R\subseteq M_2(F)$ for some field $F$.