  We prove that Galois cohomology satisfies several surprisingly strong
versions of Koszul properties, under a well known conjecture, in the finitely
generated case. In fact, these versions of Koszulity hold for all finitely
generated maximal pro-$p$ quotients of absolute Galois groups which are
currently understood. We point out several of these unconditional results which
follow from our work. We show how these enhanced versions are preserved under
certain natural operations on algebras, generalising several results that were
previously established only in the commutative case.
