I was reading the following paper: "Beilinson’s Hodge Conjecture For Smooth Varieties". They study the cycle class map $cl_{m,r}: H^{2r-m}_{\mathcal{M}}(U, \mathbb{Q}(r))\rightarrow \text{hom}_{MHS}(\mathbb{Q}(0), H^{2r-m}(U, \mathbb{Q}(r)))=\Gamma(H^{2r-m}(U,\mathbb{Q}(r)))$. The left side is the motivic cohomology and the right side is the hom in the category of mixed Hodge structures. On the second page the state the following three conjectures:

(S1) $cl_{r,m}:H^{2r-m}_{\mathcal{M}}(X, \mathbb{Q}(r))\rightarrow \Gamma(H^{2r-m}(X, \mathbb{Q}(r)))$ is surjective for smooth complex projective varieties $X$.

(S2) $cl_{r,m}:H^{2r-m}_{\mathcal{M}}(U, \mathbb{Q}(r))\rightarrow \Gamma(H^{2r-m}(U, \mathbb{Q}(r)))$ is surjective for smooth complex quasi-projective varieties $U$.

(S3) $cl_{r,m}:H^{2r-m}_{\mathcal{M}}(\mathbb{C}(X), \mathbb{Q}(r))\rightarrow \Gamma(H^{2r-m}(\mathbb{C}(X), \mathbb{Q}(r)))$ is surjective for function fields $\mathbb{C}(X)$ of complex varieties $X$.

Now for $m=0$ they claim that all three are equivalent for all $r\geq 0$. The direction going from (S1) to (S2), follows from localization sequence and using the fact that $\Gamma(H^{2r}(X, \mathbb{Q}(r)))\rightarrow \Gamma(H^{2r}(U, \mathbb{Q}(r)))$ is surjective. Going from (S2) to (S3) is just a limiting argument. I was not able to figure out how to go from (S1) to (S3). I'd appreciate if anyone can give a brief overview or point to the right source.

Another similar question one can ask is whether something similar holds for Tate conjecture? (the reduction to the generic point)

Couple of points: Note that $H_{\mathcal{M}}^{2r}(\mathcal{C}(X), \mathbb{Q}(r))=0$, so (S3) (when $m=0$) is just equivalent to saying $\Gamma(H^{2r}(\mathbb{C}(X), \mathbb{Q}(r)))=0$. It is possible by using Gersten type resolutions to show that $\Gamma(H^{2r}(\mathbb{C}(X), \mathbb{Q}(r)))=0$ implies that $\Gamma(H^{2r}(R, \mathbb{Q}(r)))=0$ where $R$ is the spectrum of a local ring (or semi-local) of a variety. But I am not sure how one can go from local rings to arbitrary quasi-projective varieties.