Deligne's conjecture for coefficient objects

Let $X_0$ be a smooth variety over $\mathbb{F}_q$, let $\ell$ be a prime number, and let $\mathcal{E}_0$ be an absolutely irreducible $\overline{\mathbb{Q}}_{\ell}$-coefficient object whose determinant has finite order.

Deligne's conjecture. The following statements hold: $\mathcal{E}_0$ is pure of weight $0$; it is $E$-rational for some number field $E\subseteq\overline{\mathbb{Q}}_{\ell}$; it is $p$-plain; and, for every prime $\ell'$ and every inclusion $\tau:E\hookrightarrow\overline{\mathbb{Q}}_{\ell'}$, there exists an absolutely irreducible $\overline{\mathbb{Q}}_{\ell'}$-coefficient object, $E$-rational with respect to $\tau$, that is $E$-compatible with $\mathcal{E}_0$.

The conjecture extends Deligne's conjecture for lisse sheaves to general coefficient objects and includes companions at all primes, including $\ell'=p$. All parts except the companion assertion (iv') are known; the missing case is $\ell'=p$ when $X_0$ has dimension at least $2$.