The open Golod–Shafarevich subgroup conjecture for pro-p extensions

Let $p$ be a prime, let $K$ be a number field, and let $L/K$ be an infinite pro-$p$ extension ramified at finitely many primes, none above $p$. A pro-$p$ group is Golod–Shafarevich if it satisfies the strict Golod–Shafarevich inequality in the sense defined in the source.

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

This is a strengthening of the non-strict virtually Golod–Shafarevich conjecture and is introduced because the strict inequality avoids the flaw discussed in the source. Its status is open.