On universal norm elements and a problem of Coleman

Suppose that $\bigcup_{n \ge 0} k_n$ is the cyclotomic $\mathbb{Z}_p$-extension of a number field $k$. In 1985, R. Coleman asked whether the quotient of the group $ ( \bigcap_{n\ge 0} N_{k_n/k} k_n^\times) \cap U_k$ (the group of units of $k$ lying in $N_{k_n/k} k_n^\times$ for all $n$, where $N_{k_n/k}$ is the norm mapping and $k_n$ is an intermediate field) over the group of universal norm units $\bigcap_{n\ge 0} N_{k_n/k}U_n$, where $U_n$ is the unit group of $k_n$, is finite. We discuss Coleman's problem for both the global units and the $p$-units, using an interpretation of the Kuz'min–Gross conjecture. Coleman claims that the quotient is finite modulo Leopoldt's conjecture and Kuz'min–Gross' conjecture under a mild condition. In this paper we improve Coleman's claim by proving the claim modulo only Kuz'min–Gross' conjecture without Leopoldt's conjecture under the same mild condition.