Let $w$ be a word of a free group. Assume that $H/\overline{w(H)}$ is virtually nilpotent for every finitely generated pro-$p$ group $H$. Let $U$ be a finitely generated free group and $T$ the maximal residually nilpotent quotient of $U/w(U)$.
If $\widehat{T_p}$ is the pro-$p$ completion of $T$, how can I show that $\widehat{T_p}$ is virtually nilpotent?
My attempts was to try to show that $\widehat{T_p}$ is isomorphic to $\widehat{U_p}/\overline{w(\widehat{U_p})}$ or something like that but I could not conclude it. I'm not looking for a proof, I just would like to know if this way works (and so, some guidance) or if there is a better approach. Thanks in the advance!
This claim [theorem 5.2] is from the paper "On the verbal width of finitely generated pro-p groups" by Andrei Jaikin-Zapirain (link at ProjectEuclid).
