The statement you want follows fairly straightforwardly from Bass' conjecture -- sufficiently straightforwardly that it may well not have a separate name of its own.

If $\Sigma$ is a sufficiently large finite set of primes, then $X$ will admit a smooth model $\mathfrak{X}$ over $\mathcal{O}_{K, \Sigma}$. Since $\mathfrak{X}$ is a finite-type $\mathbf{Z}$-scheme, Bass' conjecture implies that all Chow groups of $\mathfrak{X}$ are finite.

So it suffices to check that the natural map $CH^i(\mathfrak{X}) \to CH^i(X)$ is surjective, which is easy, because any codimension $i$ cycle on $X$ has a scheme-theoretic closure which is a codimension $i$ cycle on $\mathfrak{X}$.

This doesn't work for motivic cohomology in other degrees, incidentally (already $H^1(\operatorname{Spec} \mathbf{Q}, \mathbf{Q}(1)) = \mathbf{Q}^\times \otimes \mathbf{Q}$ has countably infinite dimension).