Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the natural morphisms

$$\mathrm{H}^i(\pi_1(X,\overline{x}),\mathscr{F}_x) \to \mathrm{H}^i_\text{ét}(X,\mathscr{F})$$

are isomorphisms for all $i\geq 0$. (See section 4 on P. Achinger's Wild Ramification and $K(\pi,1)$ spaces for a quick review on this.)

There's a natural *de Rham* analog of this story. Let $X$ be a smooth algebraic variety over a characteristic zero field $k$. The category $\textsf{DE}(X/k)$ of vector bundles on $X$ with integrable connection is tannakian. In particular, a fiber functor $\omega$ induces an equivalence of categories $\omega:\textsf{DE}(X/k)\to \textsf{Rep}_\text{fd}(\Gamma)$ for some linear algebraic group $\Gamma$. (The index fd means finite-dimensional representations.) Then, we say that $X$ is a (de Rham) $K(\pi,1)$ if for every vector bundle with integrable connection $\mathscr{E}$ and every fiber functor $\omega$ we have an isomorphism
$$\mathrm{H}^i(\Gamma,\omega(\mathscr{E}))\simeq\mathrm{H}^i_\text{dR}(X,\mathscr{E}),$$
where the cohomology on the left is algebraic group cohomology (as in chapter 15 of Milne's book Algebraic Groups), for all $i\geq 0$.

I can prove that (as in the étale context) the isomorphism above always holds for $i=0$ but I know nothing more about this story. What are some classes of algebraic varieties that are de Rham $K(\pi,1)$'s? Is a variety an étale $K(\pi,1)$ if and only if it's a de Rham $K(\pi,1)$? (Or perhaps there's only one implication?)

**In general, what's known about de Rham $K(\pi,1)$'s?**