  This paper is a greatly expanded version of Section 9.11 in arXiv:1006.4343.
A series of definitions and results illustrating the thesis in the title (where
quasi-formality means vanishing of a certain kind of Massey multiplications in
the cohomology) is presented. In particular, we include a categorical
interpretation of the "Koszulity implies $K(\pi,1)$" claim, discuss the
differences between two versions of Massey operations, and apply the derived
nonhomogeneous Koszul duality theory in order to deduce the main theorem. In
the end we demonstrate a counterexample providing a negative answer to a
question of Hopkins and Wickelgren about formality of the cochain DG-algebras
of absolute Galois groups, thus showing that quasi-formality cannot be
strengthened to formality in the title assertion.

K-Theory and Homology;Algebraic Topology;Category Theory;Rings and Algebras