Comparing direct limit and inverse limit of even $K$-groups in multiple $\mathbb{Z}_p$-extensions

Iwasawa first established a duality relating the direct limit and the inverse limit of class groups in a $\mathbb{Z}_p$-extension, and this result has recently been extended to multiple $\mathbb{Z}_p$-extensions by many authors. In this paper, we establish an analogous duality for the direct limit and the inverse limit of higher even $K$-groups in a $\mathbb{Z}_p^d$-extension.