The universal covering conjecture for isotropic semisimple groups

Let $G$ be an ${\mathbb A}^1$-connected, isotropic, semisimple, almost simple, simply connected algebraic group over $k$, split by a finite separable field extension $F/k$. Let ${\mathbf K}^{\rm M}_2$ and ${\mathbf K}^{\rm MW}_2$ denote Milnor and Milnor–Witt $K$-theory sheaves, respectively, and let $S_\alpha$ be the subgroup associated with a relative coroot $\alpha$. Universal covering conjecture. If $G$ is not of symplectic type, then the universal ${\mathbf K}^{\rm M}_2$-torsor on $G$ constructed by Brylinski–Deligne gives the universal covering of $G$ in ${\mathbb A}^1$-homotopy theory. If $G$ is of symplectic type, then there exists a ${\mathbf K}^{\rm MW}_2$-torsor on $G$ giving the ${\mathbb A}^1$-universal covering. More precisely, if the relative root system of $G$ admits a coroot $\alpha$ that becomes a coroot of small length in the root system of $G_F$, then the inclusion $S_\alpha\subset G$ induces an isomorphism with ${\bm \pi}^{{\mathbb A}^1}_1(G)$ from ${\mathbf K}^{\rm MW}_2$ in the symplectic case and from ${\mathbf K}^{\rm M}_2$ otherwise. This conjecture proposes an explicit description of the ${\mathbb A}^1$-universal cover and fundamental group for a broad class of non-split semisimple groups; its status is not resolved in the supplied source.