Fundamental group schemes of globally F-regular varieties

We prove that every tower of connected quasi-torsor covers of a connected normal globally $F$-regular projective variety over an algebraically closed field of positive characteristic stabilizes, and that the Nori fundamental group scheme of its regular locus is finite and linearly reductive. The linear reductivity asserts that any quasi-torsor cover is tame, while the finiteness is a positive-characteristic analogue of theorems of Xu and Braun on the finiteness of the fundamental group of the smooth locus of a log Fano complex projective variety.