  Let $K / \mathbb{Q}_p$ be a finite unramified extension, and let
$\mathcal{X}_n$ denote the Emerton-Gee stack parametrizing \'etale
$(\varphi,\Gamma_K)$-modules of rank $n$. It is known since the work of
Emerton-Gee that the irreducible components of the reduced special fiber of
$\mathcal{X}$ are labeled by Serre weights $\sigma$ of
$\operatorname{GL}_n(k)$. If such a component is denoted $\mathcal{X}(\sigma)$,
we prove that $\mathcal{O}(\mathcal{X}(\sigma)) \cong
\mathbb{F}[x_1,x_2,\dots,x_{n-1},x_n^{\pm 1}]$ when $\sigma$ is sufficiently
generic.
