A1-chain connectedness of A1-connected affine surfaces

Determine whether every A1-connected smooth affine surface has trivial A1-chain connected component sections over every finitely generated separable field extension.

Background

For smooth proper A1-connected surfaces, the paper states that the A1-chain connected component sheaf S(X) has trivial sections over every finitely generated separable field extension. For smooth affine A1-connected surfaces, only the corresponding triviality of S2(X) is known. The unresolved issue is whether the first A1-chain connected component S(X) is likewise trivial.

References

Following [23, Theorem 4.47], it can be shown that if X is a smooth, affine surface, which is also A1-connected, then S2(X)(F ) is trivial. However, we do not know such an affine surface X has also trivialS(X)(F ) (see [23, Remark 4.43]).

— Rationality of varieties and $\mathbb{A}^1$-homotopy theory  (2610.03680 - Choudhury et al., 2 Oct 2026) in Remark 3.31, p. 18