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