Boston's virtually Golod–Shafarevich conjecture

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 $p$. For a pro-$p$ group $G$, let $d(G)$ and $r(G)$ denote its generator rank and relation rank.

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

$$r(G)\leq d(G)^2/4.$$

This conjecture asks whether every infinite pro-$p$ extension of the indicated arithmetic type arises from a Golod–Shafarevich construction. The source attributes it to Boston and leaves it open.