TITLE: Can the valuative criteria be checked "on a dense open"?
QUESTION [5 upvotes]: The valuative criterion for separatedness (resp. properness) says that a noetherian scheme X is separated (resp. proper) if and only if

for any DVR R, with fraction field K,
  any map Spec(K)→X extends in at most one way (resp.
  extends uniquely) to a map
  Spec(R)→X.

If U⊆X is a dense open subscheme, is it sufficient to check the valuative criteria only in the cases where the map Spec(K)→X lands in U?
Intuitively, the valuative criteria are checking if it is possible to fill in a "missing point" on a curve on X (and if it is possible to fill it in in multiple ways). If there is a curve in Z=X\U with a missing point, it feels like it should be possible to find a curve in U that should have the same limit point. Also, I think it's common to verify that a compactified moduli space is separated by checking the valuative criterion using only families in the original moduli space (not "on the boundary").

REPLY [2 votes]: I'm probably not in the good decade to answer, but Lemma 4.1.1 in this article looks like what you want.