Mild pro-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e42" altimg="si15.svg"><mml:mi>p</mml:mi></mml:math> groups and the Koszulity conjectures

Let $p$ be a prime, and $\mathbb{F}_p$ the field with elements. We prove that if $G$ is mild pro-$p$ group quadratic $\mathbb{F}_p$-cohomology algebra $H^\bullet(G,\mathbb{F}_p)$, then algebras $H^\bullet(G,\mathbb{F}_p)$ $\mathrm{gr}\mathbb{F}_p[\![G]\!]$ - latter being induced by quotients of consecutive terms $p$-Zassenhaus filtration are both Koszul, they quadratically dual to each other. Consequently, maximal Galois mild, Positselski's Weigel's Koszulity conjectures hold true for such field.