Quotients of pro-$p$ groups linear over a complete Noetherian local ring

Question

Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $ p $. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ${\rm GL}_n(R)$ for some positive integer $n$. My question is the following:

Is the class of pro-$p$ groups linear over $R$ closed under taking quotients? That is, if $G$ is a pro-$p$ group which is linear over $R$ and $\pi:G\twoheadrightarrow H$ a continuous surjective homomorphism, then is $H$ also linear over $R$?

It's true for $R=\mathcal{O}_L$ where $\mathcal{O}_L$ is the ring of integers for some finite extension $L/\mathbb{Q}_p$, cf. Analytic pro-p groups. Any comments and references will be appreciated.

Answer 1

No. Let $R$ be $\mathbf{Z}_p[\![t]\!]$, take $G$ to be the Heisenberg group over $R$ (upper triangular matrices of size $3\times 3$ with diagonal 1).

The map $e_{13}:r\mapsto e_{13}(r)=1+rE_{13}$ is a group isomorphism onto the center of $G$. But $G/e_{13}(pR)$ is not linear over $R$, because it has an infinite abelian $p$-torsion subgroup and hence can't be linear (as an abstract group) over a field of characteristic zero (such as the ring of fractions of $R$).