Tame Fontaine–Mazur conjecture

Let $p$ be a prime, let $K$ be a number field, and let $S$ be a finite set of places of $K$ not dividing $p$. Let $K_S$ be the maximal pro-$p$ extension of $K$ unramified outside $S$, and write $G_S:=\operatorname{Gal}(K_S/K)$. Tame Fontaine–Mazur conjecture. For every positive integer $n$, every continuous representation $\rho:G_S\to GL_n(\mathbb{Q}_p)$ has finite image. This is the finitely and tamely ramified analogue of the weak Fontaine–Mazur conjecture. The source notes that the one-dimensional case follows from class field theory, while the higher-dimensional case remains open in general.