Positselski's pro-$\ell$ Bogomolov conjecture

Let $\ell$ be a prime and let $\mathbb{K}$ be a field containing a primitive $\ell$-th root of unity, and also $\sqrt{-1}$ if $\ell=2$. Set

$$\sqrt[\ell^\infty]{\mathbb{K}}=\mathbb{K}\left(\sqrt[\ell^n]{a},a\in\mathbb{K},n\geq1\right).$$

Positselski's pro-$\ell$ Bogomolov conjecture. The maximal pro-$\ell$ Galois group of $\sqrt[\ell^\infty]{\mathbb{K}}$ is a free pro-$\ell$ group.

This is Positselski's pro-$\ell$ version of Bogomolov's conjecture for maximal pro-$\ell$ Galois groups.