Is a Noetherian sheaf of rings stalkwise Noetherian?

Question

Let $X$ be a scheme such that $O_X$ is a coherent $O_X$-module. Assume that for every open subset $U\subset X$ and every family of coherent ideal sheaves $\{I_i\}_i$ of $O_U$, $\sum_iI_i$ is a coherent $O_U$-module. Do we have that $O_{X,x}$ is a Noetherian ring for every $x\in X$? The question is inspired by Def. A.7 in $D$-modules and Microlocal Calculus by Kashiwara.