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. Minac et al. in the case of maximal pro-$p$ Galois groups of fields with at most 2 defining relations.