Azumaya algebras over unramified extensions of function fields

Let X be a smooth variety over a field K with function field $K(X)$. Using the interpretation of the torsion part of the \'etale cohomology group H\'et²(K(X), Gₘ) in terms of Milnor-Quillen algebraic K-group K₂(K(X)), we prove that under mild conditions on the norm maps along unramified extensions of $K(X)$ over X, there exist cohomological Brauer classes in H\'et²(X, Gₘ) that are representable by Azumaya algebras on X. Theses conditions are almost satisfied in the case of number fields, providing then, a partial answer on a question of Grothendieck.