Quadratic-dual and Koszulity conjectures for maximal pro-p Galois groups

Let $p$ be a prime and let $\mathbb{K}$ be a field containing a root of unity of order $p$. For a pro-$p$ group $G$, let $\mathfrak{L}(G)$ be the restricted Lie algebra induced by its $p$-Zassenhaus filtration, and let $\mathcal{U}_{\mathfrak{L}(G)}$ be its restricted universal enveloping algebra. Write $G_{\mathbb{K}}(p)$ for the maximal pro-$p$ Galois group of $\mathbb{K}$, and let $H^\bullet(G_{\mathbb{K}}(p),\mathbb{F}_p)$ denote its $\mathbb{F}_p$-cohomology algebra. Quadratic-dual and Koszulity conjectures. The following assertions are conjectured: (i) $\mathcal{U}_{\mathfrak{L}(G_{\mathbb{K}}(p))}$ is quadratic and isomorphic to the quadratic dual of $H^\bullet(G_{\mathbb{K}}(p),\mathbb{F}_p)$; and (ii) $\mathcal{U}_{\mathfrak{L}(G_{\mathbb{K}}(p))}$ is Koszul. These conjectures relate the quadratic dual of Galois cohomology to the $p$-Zassenhaus filtration and strengthen the structural consequences expected from Bloch–Kato; their status is not specified in the source.