Motivic Cohomology and K-groups of varieties over higher local fields

For quasi-projective varieties over a higher local field $k_N$, we prove that its $K$-groups, above a suitable degree, are divisible-by-finite. We also prove the finiteness of the prime-to-$p$ torsion subgroup of certain higher Chow groups for smooth projective varieties over such fields, where $p$ denotes the final residue characteristic of $k_N$. As an application, we show that the kernel of the tame reciprocity map is uniquely $p^{\mathrm{\prime }}$-divisible. A key ingredient in achieving these results is the finiteness of étale cohomology groups over such fields.