On the purity conjecture of Nisnevich for torsors under reductive group schemes

Let R be a regular semilocal integral domain. Let G be a reductive
group scheme over R. A famous conjecture of Grothendieck and Serre
predicts that a G-torsor (a.k.a. a principal G-bundle) over R is
trivial, provided it is rationally trivial. After briefly discussing
the conjecture and its state, I'll talk about the following
generatlization: let R and G be as above, let f be an element of R
that does not belong to the square of any maximal ideal of R
(equivalently, the hypersurface {f=0} is regular). Then a G-torsor
over the localization R_f is trivial, provided that it is rationally
trivial. I will only assume basic knowledge of algebraic geometry and minimal
familiarity with torsors.