The companion conjecture for algebraic coefficient objects

Let $X$ be a smooth scheme over a finite field $k$ of characteristic $p$. An $\ell$-adic coefficient object on $X$ is algebraic if its Frobenius characteristic polynomials at all closed points have coefficients in a number field. An $\ell'$-adic coefficient object is a companion if its Frobenius characteristic polynomials coincide with those of the $\ell$-adic object at every closed point.

Companion conjecture. Any algebraic $\ell$-adic coefficient object on $X$ admits an $\ell'$-adic companion.

This is presented as equivalent to part (vi) of Deligne's conjecture and concerns the existence of companions, including crystalline companions when $\ell'=p$.