Kimura's finite-dimensionality conjecture for smooth projective varieties

Let $X$ be a smooth projective variety over $\mathbb{C}$. Its Chow motive $h(X)$ is an object of the contravariant category of Chow motives over $\mathbb{C}$, and $A^r_{\rm num}(X)$ denotes the subgroup of numerically trivial codimension-$r$ cycles.

Kimura's conjecture. All smooth projective varieties have finite-dimensional motive.

Finite-dimensionality would imply strong nilpotence properties for numerically trivial correspondences and has consequences for the structure of algebraic cycles. The conjecture is open in general, although it is known for many classes of varieties, including the cubics studied in the paper.