The reduced Boston conjecture for simple-type pro-p groups

Let $K$ be a number field and let $p$ be a prime number. A pro-$p$ group is uniform if it is powerful and torsion-free; it is hereditarily just-infinite if every open subgroup is just-infinite. A group is of simple type in characteristic zero (respectively, characteristic $p$) if it is isomorphic to a compact open subgroup of $\mathcal{G}(F)$, where $\mathcal{G}$ is a connected, simply connected, absolutely simple algebraic group over a non-Archimedean local field $F$ of characteristic zero (respectively, characteristic $p$).

Reduced Boston conjecture.

1. If $G$ is a uniform $p$-adic analytic pro-$p$ group of simple type in characteristic zero and is hereditarily just-infinite, then there is no pro-$p$ extension $L$ of $K$, ramified at finitely many primes none above $p$, with $\operatorname{Gal}(L/K)\cong G$.
2. If $G$ is an $\mathbb{F}_{p}[[T]]$-linear pro-$p$ group of simple type in characteristic $p$ and is hereditarily just-infinite, then there is no unramified pro-$p$ extension $L$ of $K$ with $\operatorname{Gal}(L/K)\cong G$.

The paper proves that this reduced conjecture is equivalent to Boston's broader unramified Fontaine–Mazur conjecture. It isolates the simple-type cases that suffice for the reduction; the source does not report a resolution.