  Let $p$ be a prime number and let ${K}$ be a field containing a root of 1 of
order $p$. If the absolute Galois group $G_{K}$ satisfies $\dim
H^1(G_{K},\mathbb{F}_p)<\infty$ and $\dim H^2(G_{K},\mathbb{F}_p)=1$, we show
that L.~Positselski's and T.~Weigel's Koszulity conjectures are true for ${K}$.
Also, under the above hypothesis we show that the $\mathbb{F}_p$-cohomology
algebra of $G_{K}$ is the quadratic dual of the graded algebra
$\mathrm{gr}_\bullet\mathbb{F}_p[G_{K}]$, induced by the powers of the
augmentation ideal of the group algebra $\mathbb{F}_p[G_{K}]$, and these two
algebras decompose as products of elementary quadratic algebras. Finally, we
propose a refinement of the Koszulity conjectures, analogous to I. Efrat's
Elementary Type Conjecture.
