On the triviality of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions

For a number field $k$ and an odd prime number $p$, we consider the maximal multiple $\mathbb{Z}_p$-extension $\tilde{k}$ of $k$ and the unramified Iwasawa module $X(\tilde{k})$, which is the Galois group of the maximal unramified abelian $p$-extension of $\tilde{k}$. In this article, we classify th