We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory *with finite coefficient $\mathbb{Z}/n$, $n$ invertible in $k$,* agree in sufficiently high degree.

Could someone please describe what is known and what is conjectured about the relation between $K_*^{et}(X)_{\mathbb{Q}}$ and $K_*(X)_{\mathbb{Q}}$?