Beilinson–Lichtenbaum generalized Hilbert 90 conjecture in Lichtenbaum motivic cohomology

Let $k$ be a field. For an abelian group $A$, define the Lichtenbaum motivic cohomology groups by

$$H^{p,q}_L(X,A):={\bf H}^p_{et}(X,A\otimes {\bf Z}(q)).$$

Beilinson–Lichtenbaum generalized Hilbert 90 conjecture. For every $n\ge 0$, one has

$$H^{n+1,n}_L(\operatorname{Spec}(k),{\bf Z})=0.$$

This is the Lichtenbaum-cohomology formulation of the generalized Hilbert 90 property and is a restatement of the earlier motivic-cohomology conjecture, so it is recorded here as the fullest formulation rather than as a separate claim.