Steinberg's representation-theoretic conjugacy conjecture

Let $G$ be an algebraic group, let $x,y\in G$, and let $\rho$ range over the irreducible rational representations of $G$. Steinberg's conjecture. The elements $x$ and $y$ are conjugate in $G$ if $\rho(x)$ and $\rho(y)$ are similar for every irreducible rational representation $\rho$ of $G$. The paper cites this as a related conjecture raised by Steinberg; its resolution is not given in the supplied text.