Topological $K$-groups of two-dimensional local fields

We consider a complete two-dimensional local field $K$ of mixed characteristic with finite second residue field and suppose that there exists a completely ramified extension $L$ of $K$ such that $L$ is a standard field. We prove that the rank of the quotient $U(1)K_2^{\mathrm{top}}K/T_K$, where $T_K$ is the closure of the torsion subgroup, is equal to the degree of the constant subfield of $K$ over $\mathbb Q_p$. I. B. Zhukov constructed a set of generators of this quotient in the case where $K$ is a standard field. In this paper, we consider two natural generalizations of this set and prove that one of them generates the whole group and the other generates its subgroup of finite index.