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

Pro-p groups with few relations and Universal Koszulity