Crystallinity conjecture for the local Galois representation attached to an unramified automorphic representation

Let $F$ be the relevant number field, let $pi$ be an automorphic representation, let $p$ be a prime of $F$ above $p$, and let $G_{p}$ denote the corresponding decomposition group. Let $\rho_{pi}$ be the Galois representation attached to $pi$.

Crystallinity conjecture. If $pi$ is unramified at $p$, then

$$\rho_{pi}|_{G_{p}}$$

is crystalline.

This stronger local property is needed in the inert-prime case and is relevant to the study of the arithmetic of automorphic Galois representations and the Bloch–Kato conjecture. The supplied text does not state whether this conjecture has been proved, so its status remains open.