Let $\mathbb{F}_{q}$ be a finite field of order $q$, and $\mathbb{F}_{q}[[T]]$ be the ring of formal power series over $\mathbb{F}_{q}$. We say that a profinite group $G$ is Noetherian if any closed subgroup of $G$ is topologically finitely generated. My question is the following:
Let $G$ be a closed subgroup of ${\rm GL}_{n}(\mathbb{F}_{q}[[T]])$ where $n\geq 2$. Suppose that $G$ is infinite and its residual image $G~\mod T$ in ${\rm GL}_{n}(\mathbb{F}_{q})$ contains ${\rm SL}_{n}(\mathbb{F}_{q})$. Is it true that $G$ is not Noetherian?
This question is inspired by the question Is every closed subgroup of GLn(K[[x]]). finitely generated?,
