Finitely generated normal pro-$\mathcal{C}$ subgroups in right angled Artin pro-$\mathcal{C}$ groups

Let $\mathcal{C}$ be a class of finite groups closed for subgroups, quotients groups and extensions. Let $\mathrm{\Gamma }$ be a finite simplicial graph and $G=G_{\mathrm{\Gamma }}$ be the corresponding pro-$\mathcal{C}$ RAAG. We show that if $N$ is a non-trivial finitely generated, normal, full pro-$\mathcal{C}$ subgroup of $G$ then $G\mathrm{/}N$ is finite-by-abelian. In the pro-$p$ case we show a criterion for $N$ to be of type $F{P}_n$ when $G\mathrm{/}N\simeq {\mathbb{Z}}_p$. Furthermore for $G\mathrm{/}N$ infinite abelian we show that $N$ is finitely generated if and only if every normal closed subgroup $N_0◃G$ containing $N$ with $G\mathrm{/}{N}_0\simeq {\mathbb{Z}}_p$ is finitely generated. For $G\mathrm{/}N$ infinite abelian with $N$ weakly discretely embedded in $G$ we show that $N$ is of type $F{P}_n$ if and only if every $N_0\le G$ containing $N$ with $G\mathrm{/}{N}_0\simeq {\mathbb{Z}}_p$ is of type $F{P}_n$.