Two conjectures by Gabber on Brauer and Picard groups

Question

In a paper I need to make reference to two conjectures by Gabber, from

• Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37

(see Conjectures 2 and 3, page 1975)

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.

Answer 1

Happy news: there have been some progress over the last 10 years. The hypersurface case of Conjecture 2 was proved in

• H. Dao, Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free. Compositio Mathematica, 148(1) (2012) pp. 145-152. doi:10.1112/S0010437X11005513, arXiv:1004.0471,

Česnavičius and Scholze recently settled both Conjectures:

• Kestutis Cesnavicius, Peter Scholze, Purity for flat cohomology, arXiv:1912.10932.