Say that a group has Property X if its pro-$p$-completion is finite for every prime $p$. For instance, every perfect group has Property X.

Is there a finitely generated, residually finite group $G$ such that every finitely generated subgroup of $G$ is either virtually nilpotent, or has Property X (but $G$ is not virtually nilpotent)? Furthermore, is there such $G$ with intermediate growth?

We would like to have examples where not much is known about their word growth.

Motivation: Henry Bradford and I found some lower bound on the word growth for some groups. We are looking for examples of groups for which this will be useful.

**Edit**: I changed the condition on the subgroups to subgroups who do not have polynomial word growth to avoid the lower bound $e^{\sqrt{n}}$ that follows from a result of Grigorchuk, see Theorem E.2 in https://arxiv.org/pdf/1512.07044.pdf.

**Edit**: see Groups with unknown word growth for a refinement of this question.