Jannsen's standard sign conjecture for homological motives

Let $k$ be a field, let $\mathrm{H}^*$ be a Weil cohomology theory with coefficients in a characteristic-zero field $F$, and let $M_{hom}(k)_F$ be the category of homological motives over $k$ associated with $\mathrm{H}^*$. Standard sign conjecture. For every $M\in\operatorname{Ob} M_{hom}(k)_F$, there exists a decomposition

$$M=M^+\oplus M^-$$

such that $\mathrm{H}^*(M^+)$ is concentrated in even degrees and $\mathrm{H}^*(M^-)$ is concentrated in odd degrees. This weakening of the Künneth standard conjecture is needed to modify the commutativity constraints and obtain a Tannakian category of homological motives; its general status is not resolved in the source.