Are motives of K3 surfaces of abelian type?

Question

I refer to the article of van Geemen https://arxiv.org/pdf/math/9903146. What van Geemen calls the Kuga-Satake-Hodge conjecture suggests that for a K3 surface $X$ over $\mathbb{C}$, the summand $h^2(X)$ of the motive of $X$ (with rational coefficients, modulo rational equivalence) should be a summand of the motive of its Kuga-Satake variety (though I don’t think this is an immediate corollary of that conjecture). This would imply that the motive of $X$ is of abelian type. One can ask if K3 surfaces over an arbitrary field have the same property—- over a finitely generated field it implies the Tate conjecture for K3 surfaces. Is this conjectured? In what generality is it known? I am also curious about whether some weaker statements are known: namely, is the motive of a K3 surface finite dimensional in the sense of Kimura?