Boston's unramified Fontaine–Mazur conjecture

Let $K$ be a number field and $S$ a finite set of primes of $K$ not containing any prime above $p$. Let $G_{K,S}$ be the Galois group of the maximal extension of $K$ unramified outside $S$.

Unramified Fontaine–Mazur conjecture. Every continuous homomorphism

$$\rho:G_{K,S}\to \operatorname{GL}_n(\mathbb{Q}_p)$$

has finite image.

This is presented as a special case of the Fontaine–Mazur conjecture under standard conjectures in algebraic geometry. The source reports partial results and reductions but gives no resolution of the conjecture in general.