Purity, splitness and cellularity conjectures for mixed motives

Let $H$ be a mixed tight Weil cohomology, let $\mathcal{MW}^+_{\rm pure}$ denote the category of purified mixed Weil cohomologies, and let $\Phi^+_{\rm pure}$ be the associated functor. Let $(\mathcal{MW}^+,MW^+)$ denote the category of mixed Weil cohomologies equipped with its universal cohomology. A category is split when all its extensions split, and a pair is cellular in the sense of the cellularity property described in the paper.

Purity, splitness and cellularity conjectures. The following properties should hold:

1. Purity: The functor $\Phi^+_{\rm pure}$ is a tensor equivalence.
2. Splitness: The category $\mathcal{MW}^+_{\rm pure}$ is split.
3. Cellularity: $(\mathcal{MW}^+,MW^+)$ is cellular.

These expectations would identify purified mixed Weil cohomologies with the corresponding pure theory, imply that extensions in the purified category split, and provide cellular constructions of realization functors for mixed motives. The surrounding discussion presents them as expected properties; no resolution is supplied here.