TITLE: State of the art for Gersten's conjecture for K-theory?
QUESTION [15 upvotes]: 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 $(K_n)_{Zar} = (K_n)_{Nis}$).

REPLY [3 votes]: On the arXiv there is a paper of Satoshi Mochizuki on "Gersten’s conjecture for commutative discrete valuation rings" from the year 2007. In this paper Mochizuki claims to prove the Gersten conjecture for all commutative DVRs. But the paper didn't appear in any journal, apperently. Does anyone know if it has been verified?