Twisted conjugacy classes in Chevalley groups

Let $G$ be a group and $\varphi\colon G\to G$ its automorphism. We say that elements $x$ and $y$ of $G$ are twisted $\varphi$-conjugate, or merely $\varphi$-conjugate (written $x\sim_\varphi y$), if there exists an element $z$ of $G$ for which $x=zy\varphi(z^{-1})$. If, in addition, $\varphi$ is an identical automorphism, then we speak of conjugacy. The $\varphi$-conjugacy class of an element $x$ is denoted by $[x]_\varphi$. The number $R(\varphi)$ of these classes is called the Reidemeister number of an automorphism $\varphi$. A group is said to possess the $R_\infty$ property if the number $R(\varphi)$ is infinite for every automorphism $\varphi$.
We consider Chevalley groups over fields. In particular, it is proved that if an algebraically closed field $F$ of characteristic zero has finite transcendence degree over $\mathbb Q$, then a Chevalley group over $F$ possesses the $R_\infty$ property. Furthermore, a Chevalley group over a field $F$ of characteristic zero has the $R_\infty$ property if $F$ has a periodic automorphism group. The condition that $F$ is of characteristic zero cannot be discarded. This follows from Steinberg's result which says that for connected linear algebraic groups over an algebraically closed field of characteristic zero, there always exists an automorphism $\varphi$ for which $R(\varphi)=1$.