Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group schemes over $k$}\}$ and $\mathsf{A}_k=\{\text{abelian varieties over $k$ modulo isogeny}\}$. The category $\mathsf{G}_k$ is abelian (see this answer), and so is $\mathsf{A}_k$, though I don't know of a nice reference for this. Two questions:

Is $K_0(\mathsf{A}_k)$ generated by Jacobians of curves? (This question was asked by Grothendieck quite a while ago - I am wondering if any progress has been made).

Is anything at all known about $\mathsf{G}_k$? (I acknowledge that this is a very vague question - more of a reference request)