Stability of sheaves of locally closed and exact forms

For any smooth projective variety $X$ of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$ with $\mu(\Omega^1_X)>0$. If ${\rm T}^{\ell}(\Omega^1_X)$ ($0<\ell<n(p-1)$) are semi-stable, then the sheaf $B^1_X$ of exact 1-forms is stable. When $X$ is a surface with $\mu(\Omega^1_X)>0$ and $\Omega^1_X$ is semi-stable, the sheaf $B^2_X$ of exact 2-forms is also stable. Moreover, under the same condition, the sheaf $Z^1_X$ of closed 1-forms is stable when $p>3$, and $Z^1_X$ is semi-stable when $p=3$.