The companions conjecture for irreducible coefficient objects

Let $X/\mathbb{F}_{q}$ be a smooth variety, and let $\mathcal{F}$ be an irreducible coefficient object on $X$ with algebraic determinant. Companions conjecture. There exists a number field $E$ such that $\mathcal{F}$ belongs to a complete $E$-compatible system. Here a complete compatible system includes the coefficient objects at all finite places, including the required crystalline objects at places above the characteristic, and is intended as a refinement of Deligne's conjecture on algebraicity and companions. The source leaves this conjecture unresolved.