@Andrei: To prove your first claim, should I check that the factors of the (topological) lower central series are (topologically) finitely generated from basic commutators, or is there a simpler argument? About the second claim, how did you choose (topological) generators of $\overline{T}$ in $T$? Sorry about these questions, I'm just beginning to learn profinite groups.

@Andrei: Many thanks! Though, I must say I need some time to fully understand your interesting example. Do you think it is possible to do something similar when the group is nilpotent? (the case I'm in fact interested in). Thanks again.

@Yiftach: Thanks for the idea and for the reference! I'm thinking about them too.

@Alves: Sure! And that was my first belief too! Unfortunately I'm having a hard time to prove this.