I have seen the claim that Beilinson Lichtenbaum implies that higher algebraic $K$ groups coincides with etale ones integrally in high enough degrees. Is this statement accurate? What conditions are required and how to derive it?
To my knowledge the most general known statement has been proven by Clausen and Mathew in their paper Hyperdescent and étale K-theory as Theorem 1.2. The precise conditions on your commutative ring (or more generally algebraic space) are a bit technical to summarize, but are very general and give explicit bounds. Away from the residue chacteristics one can apply the Voevodsky-Rost norm residue theorem and at the residue characteristics they manage to apply a reduction to topological cyclic homology ($TC$). I recommend looking at their paper for a more detailed explanation.