Weak Fontaine–Mazur conjecture for unramified pro-p extensions

Let $K$ be a number field. An unramified pro-$p$-extension is a Galois pro-$p$ extension of $K$ unramified at every place. Weak Fontaine–Mazur conjecture. Every unramified pro-$p$-extension of $K$ whose Galois group is $p$-adic analytic is finite. The source calls this elementary to state but completely out of reach; it is presented as a consequence of the Fontaine–Mazur conjecture together with Tate's conjecture.