Second A1-chain iteration for A1-connected quasi-projective surfaces

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

Background

The paper reports that triviality of S2(X)(F) is known for smooth affine A1-connected surfaces, while the analogous assertion for arbitrary smooth quasi-projective A1-connected surfaces remains unresolved. Here S2 denotes the second iteration of the A1-chain connected component construction, and F is a finitely generated separable field extension of the base field.

References

We also do not know whether an A1-connected, smooth, quasi-projective surface X always has trivial S2(X)(F ).

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