Pro-p groups with few relations and Universal Koszulity

Let $p$ be a prime. We show that if a pro-$p$ group with at most 2 defining relations has quadratic ${\mathbb{F}}_p$-cohomology, then such algebra is universally Koszul. This proves the “Universal Koszulity Conjecture” formulated by J. Min'a\v{c} et al. in the case of maximal pro-$p$ Galois groups of fields with at most 2 defining relations.