# Infinite pro-$p$ group of finite solvable length and finite coclass

I was reading about infinite pro-$$p$$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. I asked this question in math.stackExchange before posting it here; there were no answers there, I thought I might share it in mathoverflow. My advance apologies if anything is inappropriate.

I was thinking about solvability and I think it can be shown that if $$G$$ is a solvable group of solvable length $$l$$ then every subgroup and quotient of $$G$$ has solvable length at most $$l$$ (please correct me if I am wrong). My question is related to the "opposite" of this property.

My question is

Let $$S$$ be an infinite pro-$$p$$ group of finite coclass. Suppose there exists a non-negative integer $$t$$ such that the solvable length of each lower central series quotient $$S/\gamma_i(S)$$ is less than or equal to $$l$$ for all $$i\ge t$$. Then is it true that the $$S$$ is solvable with solvable length less than or equal to $$l$$?

To recall, the coclass of a finite $$p$$-group $$G$$ of order $$p^n$$ is defined as $$n-c$$ where $$c$$ is the nilpotency class of $$G$$. In case of infinite pro-$$p$$ groups, an infinite pro-$$p$$ group $$S$$ is said to be of finite coclass $$r$$ if its lower central series quotients $$S/\gamma_i(S)$$ are finite $$p$$-groups and $$S/\gamma_i(S)$$ has coclass $$r$$ for all $$i\ge t$$ for some $$t\ge 0$$.

The answer is yes, $$S$$ is an inverse limit of its lower central quotients. As these have bounded derived length, the same goes for the Cartesian product of these groups.
By the way, all pro-$$p$$ groups of finite coclass are solvable, that's Theorem D of the coclass theory.