Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence

$$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \bigoplus_{x \in X^{(1)}} K_{n - 1}(k(x))$$

where $\mathcal{K}_n$ is the Zariski sheaf associated to $K_n$?

More specifically, I would like the following to be true.

1) The above sequence is exact when $X$ is a (separated noetherian) regular scheme of dimension one.

2) $\mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x))$ is injective for all regular (separated noetherian) schemes.

And I would like these to be true with the Nisnevich sheafification of $K_n$ as opposed to the Zariski one (I'm aware that in many cases the two sheafifications are the same so I guess another part of the question is in which cases $(K_n)_{Zar} = (K_n)_{Nis}$).