Tight closure/Plus closure/Abelsche Varietät/Problem/en

Let ${}Y$ be an abelian variety in positive characteristic ${}p$ and let ${}c\in H^{1}(Y,{\mathcal {S}})$ be a cohomology class of a vector bundle such that the cohomological dimension of the corresponding torsor equals the dimension of ${}Y$. Is it true that ${}c$ can be annihilated by an iteration of the ${}[p]$-multiplication?