Fontaine–Mazur unramified conjecture for representations

Let $K$ be a number field, and let

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

denote the Galois group of the maximal unramified extension of $K$. Fontaine–Mazur's unramified conjecture. Every continuous Galois representation

$$\rho:G^{ur}_K\longrightarrow \operatorname{GL}_m(\mathbb{Z}_p)$$

has finite image. This is the unramified form of the Fontaine–Mazur conjecture. The paper studies it through $p$-adic analytic quotients and establishes new special cases, while the general assertion remains open.