$E_2$ formality via obstruction theory

I will explain how to approach the question of $E_2$-formality of differential graded algebras over a prime field via obstruction theory. In particular, $E_2$-algebras whose cohomology ring is a polynomial algebra on even degree classes are intrinsically formal. As a consequence we can prove $E_2$-formality of the classifying space of some compact Lie group or of Davis-Januszkiewicz spaces.