Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). Suppose that $T$ extends along codimension $1$ points of $\mathbb{A}^n_k$, i.e. the torsor class $[T_{k(\mathbb{A}^n)}] \in \mathrm{H}^1(k(\mathbb{A}^n),G)$ lies in the image of $\mathrm{H}^1(\mathcal{O}_{\mathbb{A}^n,Z},G)\rightarrow \mathrm{H}^1(k(\mathbb{A}^n),G)$ for all closed irreducible $Z\subset \mathbb{A}^n$ of codimension $1$. (This is true, for example, if $\mathbb{A}^n\setminus V$ has codimension $\geq 2$ in $\mathbb{A}^n$.)

Question: does there exist a $G$-torsor $T'$ on $\mathbb{A}^n$ such that $T$ and $T'|_V$ are locally isomorphic in the Zariski topology?

Some remarks:

- By known cases of the Grothendieck-Serre conjecture [Panin], it suffices to show that there exists a $G$-torsor $T'$ on $\mathbb{A}^n$ with $[T_{k(\mathbb{A}^n)}]=[T'_{k(\mathbb{A}^n)}]$ in $\mathrm{H}^1(k(\mathbb{A}^n),G)$.
- We cannot require that $T$ literally extends to a $G$-torsor on $\mathbb{A}^n$; indeed there exists a $\mathrm{GL}_2$-torsor on $\mathbb{A}^3\setminus\{0\}$ that does not extend to $\mathbb{A}^3$.
- The question has a negative answer if we allow $V$ to be an open subset in an arbitrary smooth affine $k$-variety [Antieau--Williams].
- The question is trivial for $G = \mathrm{GL}_n$ or $\mathrm{Sp}_n$, since every torsor under such a group is Zariski locally trivial.

*Panin, Ivan A.*, **Proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing a field**, Izv. Math. 84, No. 4, 780-795 (2020); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 84, No. 4, 169-186 (2020). ZBL1458.14024.

*Antieau, Benjamin; Williams, Ben*, **Topology and purity for torsors**, Doc. Math. 20, 333-355 (2015). ZBL1349.14068.