A conjecture of Nisnevich predicts that for a smooth variety $X$ over a field, a smooth divisor $D$ in $X$, and a totally isotropic reductive $X$-group scheme $G$, every generically trivial $G$-torsor on $X \setminus D$ trivializes Zariski locally on $X$. I will discuss this conjecture and related questions about torsors under reductive groups over regular rings.