Uniform weak Fontaine–Mazur conjecture

Let $K$ be a number field and let $L/K$ be an everywhere unramified Galois pro-$p$ extension. A pro-$p$ group is uniform if it is a uniform pro-$p$ group in the sense of Lazard. Uniform weak Fontaine–Mazur conjecture. There is no number field $K$ for which an infinite everywhere unramified Galois pro-$p$ extension $L$ exists with $\operatorname{Gal}(L/K)$ uniform. By Lazard's theory, this is presented as a reformulation of the weak Fontaine–Mazur conjecture; the source gives no resolution.