Mild pro-p groups and the Koszulity conjectures

Let p be a prime, and Fp the field with p elements. We prove that if G is a mild pro-p group with quadratic Fp-cohomology algebra H•(G,Fp), then the algebras H•(G,Fp) and grFp[[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.