Bogomolov's cohomologically trivial factorization conjecture

Let $G={\mathfrak G}_{\ell}$ be an $\ell$-Sylow subgroup of the absolute Galois group of a function field over an algebraically closed field. Set

$$M={\rm Hom}(G^{(1)}/G^{(2)},\mathbb Z/\ell^m),$$

where $G^{(1)}=[G,G]$ and $G^{(2)}=[G^{(1)},G^{(1)}]$, and let $G^a=G/G^{(1)}$. The subgroup $M^{G^a}$ consists of the $G^a$-invariants of $M$. Bogomolov's factorization conjecture. The projection $M\to M/M^{G^a}$ factors as

$$M\hookrightarrow D\twoheadrightarrow M/M^{G^a},$$

where $D$ is a cohomologically trivial $G^a$-module. This conjecture is intended to complement the Freeness conjecture by implying the required surjectivity on cohomology and hence the description of kernels by trivial symbols. Its status is not resolved in the supplied text.