Commutator identity involving generalized derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U, C be the extended centroid of \(R,\, F\) and G be two nonzero generalized derivations of R and \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C which is not central valued on R. If $$\begin{aligned} {[}F(u)u, G(v)v]=0 \end{aligned}$$ for all \(u,v\in f(R)\), then there exist \(a,b\in U\) such that \(F(x)=ax\) and \(G(x)=bx\) for all \(x\in R\) with \([a, b]=0\) and \(f(x_1,\ldots ,x_n)^2\) is central valued on R.