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In a paper I need to make reference to 2 conjectures by Gabber

(see Conjectures 2 and 3, page 1975)

http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf

1) Let $R$ be a strictly henselian complete intersection noetherian local ring of dimension at least 4. Then $Br'(U_R) = 0$ (the cohomological Brauer group of the punctured spectrum is $0$).

2) Let $R$ be a complete intersection noetherian local ring of dimension 3. Then $Pic(U_R)$ is torsion-free.

Does anyone know of any new developments on these conjectures beyond the Oberwolfach report above? I tried MathScinet but could not find anything. May be someone in the Arithmetic Geometry community happen to know some news on these? Thanks a bunch.

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2  
Have you tried e-mailing Gabber with your question? –  Ryan Budney Jan 3 '10 at 6:13
    
I am doing it right now! Stay tuned (: –  Hailong Dao Jan 3 '10 at 6:18
    
I'd also like to know more about this. Progress on the purity conjecture for the Brauer group would be really interesting! –  Clark Barwick Jan 18 '10 at 15:26

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