Birationality and A1-invariance conjecture for A1-connected components of proper schemes

Let $k$ be a field and let $X$ be a proper scheme of finite type over $k$. Denote by $\pi_0^{{\mathbb A}^1}(X)$ the ${\mathbb A}^1$-connected components sheaf and by $\pi_0^{b{\mathbb A}^1}(X)$ the birational ${\mathbb A}^1$-connected components sheaf. There is a canonical morphism

$$\pi_0^{{\mathbb A}^1}(X)\longrightarrow\pi_0^{b{\mathbb A}^1}(X).$$

Birationality and ${\mathbb A}^1$-invariance conjecture. For any proper scheme $X$ of finite type over a field $k$, this canonical morphism is an isomorphism; equivalently, $\pi_0^{{\mathbb A}^1}(X)$ is birational and ${\mathbb A}^1$-invariant. The preceding proposition establishes only a factorization through the birational sheaf and a bijection on sections over finitely generated separable extension fields, so the sheaf-level isomorphism remains open in the supplied text.