Abelianness and strict A1-invariance of connected components of reductive groups

Let $k$ be a field and let $G$ be a reductive algebraic group over $k$. Write $pi_0^{{\mathbb A}^1}(G)$ for the sheaf of ${\mathbb A}^1$-connected components of $G$. A sheaf of groups is strictly ${\mathbb A}^1$-invariant if it is a sheaf of abelian groups and its Nisnevich cohomology presheaves are ${\mathbb A}^1$-invariant in every degree.

Abelianness and strict invariance conjecture. For every reductive group $G$ over $k$, the sheaf $\pi_0^{{\mathbb A}^1}(G)$ is a sheaf of abelian groups. Moreover, $\pi_0^{{\mathbb A}^1}(G)$ is strictly ${\mathbb A}^1$-invariant.

This would place the ${\mathbb A}^1$-connected components of every reductive group in the abelian category of strictly ${\mathbb A}^1$-invariant sheaves. The abelianness of these sheaves is stated as an open question in the source, and strict ${\mathbb A}^1$-invariance would then follow from the relevant ${\mathbb A}^1$-homotopy-theoretic results; no general proof is given here.