$1$-smooth pro-$p$ groups and Bloch–Kato pro-$p$ groups

Let $p$ be a prime. A pro-$p$ group $G$ is said to 1-smooth if it can endowed with homomorphism of groups $G\to1+p\mathbb{Z}_p$ satisfying formal version Hilbert 90. By Kummer theory, maximal Galois fields containing root 1 order $p$, together the cyclotomic character, are 1-smooth. We prove that finitely generated $p$-adic analytic if, and only occurs as field $p$. This gives positive answer De Clerq-Florence's Smoothness Conjecture - which states Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from 1-smoothness for class groups.