  We complete the proof of the Nisnevich conjecture in equal characteristic:
for a smooth algebraic variety $X$ over a field $k$, a $k$-smooth divisor $D
\subset X$, and a reductive $X$-group $G$ whose base change $G_D$ is totally
isotropic, we show that each generically trivial $G$-torsor on $X\setminus D$
trivializes Zariski semilocally on $X$. In mixed characteristic, we show the
same when $k$ is a replaced by a discrete valuation ring $\mathcal{O}$, the
divisor $D$ is the closed $\mathcal{O}$-fiber of $X$, and either $G$ is
quasi-split or $G$ is only defined over $X \setminus D$ but descends to a
quasi-split group over $\mathrm{Frac}(\mathcal{O})$ (a Kisin-Pappas type
variant). Our arguments combine Gabber-Quillen style presentation lemmas with
excision and reembedding d\'{e}vissages to reduce to analyzing generically
trivial torsors over a relative affine line. We base this analysis on the
geometry of the affine Grassmannian, and we we give a new proof for the
Bass-Quillen conjecture for reductive group torsors over $\mathbb{A}^d_R$ in
equal characteristic. As for the affine Grassmannian itself, we show that for
totally isotropic $G$ it is the presheaf quotient $LG/L^+G$.
