Unramified Fontaine–Mazur conjecture via uniform quotients

Let $K$ be a number field and $p$ a prime. Let $K^{ur}(p)$ be the maximal unramified pro-$p$ extension of $K$, and set

$$G^{ur}_K(p)=\operatorname{Gal}(K^{ur}(p)/K).$$

A quotient is uniform if it is a uniform pro-$p$ group. Unramified Fontaine–Mazur conjecture. Every uniform quotient $G$ of $G^{ur}_K(p)$ is trivial. This is presented as an equivalent formulation of the unramified Fontaine–Mazur conjecture; the paper proves special cases for certain totally imaginary, especially imaginary quadratic, fields when $p=2$.