Kimura–O'Sullivan's finite-dimensionality conjecture for motives

Let ${\mathcal M}_k$ be the category of motives over the field $k$. A motive $M\in{\mathcal M}_k$ is finite-dimensional in the sense of Kimura and O'Sullivan if it admits the corresponding finite-dimensional decomposition. Kimura–O'Sullivan's conjecture. Every motive $M\in{\mathcal M}_k$ is finite-dimensional. This conjecture extends finite-dimensionality results known for motives of products of smooth projective curves, varieties dominated by such products, and certain varieties of dimension at most three. It remains wide open.