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Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups $$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$ are finitely generated? We know that they are finite-dimensional after tensoring with $\mathbb Q$, since they become identified with the rational $K$-theory $K_{2n-1}(F)_{\mathbb Q}$ whose dimension was computed by Bloch. But are they finitely generated integrally? A reference, if it exists, would be appreciated.

Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups $$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$ are finitely generated? We know that they are finite-dimensional after tensoring with $\mathbb Q$, since they become identified with the rational $K$-theory $K_{2n-1}(F)_{\mathbb Q}$ whose dimension was computed by Borel. But are they finitely generated integrally? A reference, if it exists, would be appreciated.

Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups $$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$ are finitely generated? We know that they are finite-dimensional after tensoring with $\mathbb Q$, since they become identified with the rational $K$-theory $K_{2n-i}(F)_{\mathbb Q}$ whose dimension was computed by Bloch. But are they finitely generated integrally? A reference, if it exists, would be appreciated.

Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups $$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$ are finitely generated? We know that they are finite-dimensional after tensoring with $\mathbb Q$, since they become identified with the rational $K$-theory $K_{2n-1}(F)_{\mathbb Q}$ whose dimension was computed by Bloch. But are they finitely generated integrally? A reference, if it exists, would be appreciated.