A Beauville-Laszlo-type descent theorem for locally Noetherian schemes

Let $X$ be a locally Noetherian scheme with a closed subscheme $Z$. Let $\mathcal{X}$ be the completion of $X$ at $Z$, considered as a formal scheme. We show that a coherent sheaf on $X$ is equivalently given by a coherent sheaf on $\mathcal{X}$, a coherent sheaf on the complement of $Z$, and an isomorphism of pullbacks of these sheaves to a certain adic space $W$. By defining $W$ as an adic space instead of as a Berkovich space we are able to generalize the descent result of Ben-Bassat and Temkin from finite type $k$-schemes to locally Noetherian schemes.