Twisted conjugacy in generalized Thompson groups of type F

If $\varphi$ is an automorphism of a group $G$ and $x,y\in G$, we say that $x$ and $y$ are $\varphi$-twisted conjugates if there exists an $z\in G$ such that $y=z\mathrm{.}x\mathrm{.}\varphi (z^{-1})$. This is an equivalence relation. If there are infinitely many $\varphi$-twisted conjugacy classes for every automorphism $\varphi$ of $G$ we say that $G$ has the $R_{\mathrm{\infty }}$-property. We prove that the generalized Richard Thompson groups $F_n$ and $F(l,A,P)$ have the $R_{\mathrm{\infty }}$-property.