The main result of this article is to reduce a proof of the conjecture to a statement about principal bundles on affine line over a regular local scheme. This reduction is obtained via a theory of nice triples, which goes back to the ideas of Voevodsky. As an application, an unpublished result due to Gabber is proved.
"Nice triples and the Grothendieck–Serre conjecture concerning principal G-bundles over reductive group schemes." Duke Math. J. 168 (2) 351 - 375, 1 February 2019. https://doi.org/10.1215/00127094-2018-0042