Abstract commensurability of Gupta-Sidki group

Question

I know that all the infinite finitely generated subfroups of the Gupta-Sidki 3 group are (abstract) commensurable with G or GxG, however I have not found any proof for generalising this for any odd prime p>3, or proving it wrong with some contradiction. Does someone know if this result is also true?

Answer 1

The result does generalise. If $G$ is the Gupta-Sidki $p$-group, for $p\geq 3$, then any finitely generated subgroup of $G$ is commensurable with a direct product of a certain number, between $1$ and $p-1$, of copies of $G$. The proof of this can be found here.