Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = CH(X)\otimes_{\mathbb Z} \mathbb Q$. My question is

Suppose $CH(X)_{\mathbb Q}$ is finite-dimensional as a $\mathbb Q$-vector space. When can we conclude that $H^d(X, \mathcal O_X) =0$?

I suspect that the answer might be always yes: in dimension one, such varieties have to be rational. In dimension two, I think it follows from a paper of Mumford. Also, it is true for flag manifolds.

More vaguely, I am also quite interested in:

Suppose $CH(X)_{\mathbb Q}$ is finite-dimensional as a $\mathbb Q$-vector space. What can we say about the geometry of $X$?

Any partial answers would be appreciated. Thanks!