2 problems

Open Golod–Shafarevich subgroup conjecture. The group $\operatorname{Gal}(L/K)$ contains an open Golod–Shafarevich pro-$p$ subgroup.

Boston's virtually Golod–Shafarevich conjecture. The group $\operatorname{Gal}(L/K)$ contains an open pro-$p$ subgroup $G$ such that $d(G)$ and $r(G)$ are finite and