Finite symmetric integral tensor categories with the Chevalley property

We prove that every finite symmetric integral tensor category $mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to $text{sVec}$. This proves Ostriks conjecture cite[Conjecture 1.3]{o} in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $mathcal{G}$ over $k$ and a grouplike element $epsilonin k[mathcal{G}]$ of order $le 2$, whose action by conjugation on $mathcal{G}$ coincides with the parity automorphism of $mathcal{G}$, such that $mathcal{C}$ is symmetric tensor equivalent to $Rep(mathcal{G},epsilon)$. In particular, when $mathcal{C}$ is unipotent, the functor lands in $Vect$, so $mathcal{C}$ is symmetric tensor equivalent to $Rep(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of cite{g} to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper cite{Co}, and, more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $ e 2$ is always a Serre subcategory.