Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of topological rings for all $j$. Call $\mathfrak{X}$ the limit $\varprojlim_j \mathfrak{X}_j$.

Let $\text{Coh}(\mathfrak{X})$, $\text{Coh}(\mathfrak{X}_j)$, be the categories of topologically finitely presented $\mathcal{O}$-modules.

Do we have $2\mbox{-}\varinjlim_j\text{Coh}(\mathfrak{X}_j) = \text{Coh}(\mathfrak{X})$?

In other words, does a topologically finitely presented $\mathcal{O}_{\mathfrak{X}}$-module descend to some finite layer, and do morphisms too?

Of course, for algebraic schemes this is true and standard. A reference?