Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic zero and fix a point $x\in X(K)$.
We can associate to $X$ two Tannakian categories: the category of Higgs bundles on $X$ and the category of vector bundles with flat connection on $X$. The fiber functor in both cases is given by taking the fiber at $x$. Denote by $\pi_1^\text{Higgs}(X,x)$ and $\pi_1^\text{loc. sys.}(X,x)$ the respective pro-algebraic groups associated to these Tannakian categories.
If $K=\mathbb{C}$, then these two Tannakian categories are both equivalent to the category of representations of the topological fundamental group $\pi_1(X,x)$ by the nonabelian Hodge correspondence and the Riemann-Hilbert correspondence, respectively. Therefore, we have isomorphisms $$\pi_1^\text{Higgs}(X,x) \cong \pi_1^\text{loc. sys.}(X,x) \cong \widehat{\pi_1}(X,x)$$ where $\widehat{\pi_1}(X,x)$ denotes the pro-algebraic completion of $\pi_1(X,x)$ (and I think that these isomorphisms are canonical but I'm not totally sure). Hence we have an isomorphism between two pro-algebraic groups with a purely algebraic definition but whose construction a priori crucially uses the complex analytic topology.
If $K$ is not taken to be $\mathbb{C}$, how do the two candidates for a pro-algebraic fundamental group $\pi_1^\text{Higgs}(X,x)$ and $\pi_1^\text{loc. sys.}(X,x)$ compare? Are they still always (canonically) isomorphic?
The same question can also be asked if we replace the two Tannakian categories with their full subcategories of nilpotent objects, which corresponds to replacing the Tannakian groups with their pro-unipotent quotients. These pro-unipotent $\pi_1$'s are somewhat nicer than the pro-algebrac $\pi_1$'s because they are are compatible with base change (at least I know that this is true for the local systems $\pi_1$, I'm not totally about the Higgs $\pi_1$). I am also interested in an answer to the question above for the pro-unipotent $\pi_1$'s.
