TITLE: Embedding proper algebraic spaces
QUESTION [9 upvotes]: Does every proper algebraic space (over a field, say) admit a closed immersion into a smooth proper algebraic space?
Remark: Of course, if we say "projective" instead of "proper" then the answer is tautologically "yes":  we can take the ambient variety to be some projective n-space.  But I'm curious about the general case.

REPLY [6 votes]: I do not know the general answer to this interesting question.
However, there are surely examples of proper algebraic varieties that admit no embeddeding into smooth schemes. 
Look for instance to this paper by Roth and Vakil, page 11. 
Note that in the paper the authors claim that their example admits no embedding into a smooth algebraic space; however, their proof actually only work for smooth schemes, see the corrigendum in Vakil's webpage here.