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Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$. Let $P$ be a parabolic subgroup of $G$, $\mathscr{F}:=G/P$ the partial flag variety associated to $P$. For a $G$-variety $X$, ...

Let $R$ be a unitary associative ring. If $R$ is commutative, then one defines $SK_1(R)$ as the quotient $$SK_1(R)=SL(R)/E(R)$$ (Definition 2.8 of http://dmle.cindoc.csic.es/pdf/...