  For a prime number $\ell$ we say that an oriented pro-$\ell$ group
$(G,\theta)$ has the Bogomolov property if the kernel of the canonical
projection on its maximal $\theta$-abelian quotient $\pi^{ab}_{G,\theta}\colon
G\to G(\theta)$ is a free pro-$\ell$ group contained in the Frattini subgroup
of $G$. We show that pro-$\ell$ groups of elementary type have the Bogomolov
property. This shows that Efrat's Elementary Type Conjecture implies a positive
answer to Positselski's version of Bogomolov's Conjecture on maximal pro-$\ell$
Galois groups of a field $K$ in case that $K^\times/(K^\times)^\ell$ is finite.
Secondly, it is shown that for an $H^\bullet$-quadratic oriented pro-$\ell$
group $(G,\theta)$ the Bogomolov property can be expressed by the injectivity
of the transgression map $d_2^{2,1}$ in the Hochschild-Serre spectral sequence.
