I was reading the following paper: ["Beilinson’s Hodge Conjecture For Smooth Varieties"](https://arxiv.org/pdf/1104.4364.pdf). They stuudy 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:

1) (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 $X$.

2) (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 $U$.

3) (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.