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 finitely generated. [Edit: as user "guest" points out, this last step only works after $\otimes \mathbf{Q}$; to get finite generation with $\mathbf{Z}$ coefficients you need something slightly stronger, the "motivic Bass conjecture" which is the conjecture that finite-type $\mathbf{Z}$-schemes have finitely-generated motivic cohomology.]

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).

Height pairings between algebraic cycles(conjecture 5.0). $\endgroup$1more comment