I call a profinite group $G$ Noetherian, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is finitely generated.
A profinite group $G$ is called just-infinite if every nontrivial $M \lhd_c G$ is open.
Let $K$ be a profinite Noetherian just-infinite group. Must $K$ be the profinite completion of some residually finite group $R$?