The ^1-connectedness conjecture for reductive algebraic groups

Let $G$ be a reductive algebraic group over a field $k$, and let $F$ be a field extension of $k$. Write $G(F)/R$ for the set of $R$-equivalence classes of $F$-points of $G$, and let $\pi_0^{{\mathbb A}^1}(G)(F)$ denote the zeroth ${\mathbb A}^1$-homotopy sheaf of $G$ evaluated at $F$.

The conjecture. For every field extension $F$ of $k$,

$$\pi_0^{{\mathbb A}^1}(G)(F)=G(F)/R.$$

This conjecture asks whether $R$-equivalence completely describes ${\mathbb A}^1$-connected components of reductive algebraic groups. It extends the comparison proved in the paper for anisotropic groups, while the general case remains open.