The weak unramified Fontaine–Mazur conjecture

Let $K$ be a number field, and let $G_{K,\emptyset}(p)$ be the Galois group of the maximal everywhere unramified pro-$p$-extension of $K$.

Weak unramified Fontaine–Mazur conjecture. Every continuous homomorphism

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

has finite image.

This is implied by the unramified Fontaine–Mazur conjecture for $S=\emptyset$. Although $G_{K,\emptyset}(p)$ can be infinite by the Golod–Shafarevich theorem, the conjecture asserts that it has no infinite-image $p$-adic linear representations. The source gives no resolution in general.