Let $k$ be a field. In what generality is it true that higher etale homotopy groups of $\mathrm{Spec}\,k$ vanish?

If the absolute Galois group is finite, we have a universal cover $\mathrm{Spec}\,k^{sep}\rightarrow\mathrm{Spec}\,k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $\pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.

I am not sure what happens for general fields.