Gieseker conjecture for homogeneous spaces

We prove Gieseker conjecture for an homogeneous space $X$, saying that if $X$ has no non-trivial tame coverings then it has no non-trivial regular singular $\mathscr{O}_X$-coherent $\mathscr{D}_{X/k}$-modules. In order to do so we prove a K\"unneth formula for the regular singular stratified fundamental group and a base change for Gauss-Manin stratifications in the non-proper case.