Let $X$ be a smooth scheme over $\mathbb{F}_{p}$ for a prime number $p$. As far as I understand, there is a surjective morphism from $\Omega^\bullet_{W\mathcal{O}_X} \to W \Omega_{X}^\bullet$ which induces an isomorphism $\Omega^\bullet_{W\mathcal{O}_X}/(T+Fil^n \Omega^\bullet_{W\mathcal{O}_X}) \to W_{n}\Omega_X^\bullet$ where $T$ is the graded differential ideal consisting of $p-$torsion elements of $\Omega^\bullet_{W\mathcal{O}_X} $ and $Fil^n$ is the kernel of the natural projection map from $\Omega^\bullet_{W \mathcal{O}_X} \to \Omega^\bullet_{W_n\mathcal{O}_X} $. A reference for this is "Complexe de de-Rham Witt et Cohomologie Cristalline" by Luc Illusie.

The question is: Suppose there exists an integer $N$ such that $H^i(W_{n}\Omega_{X}^\bullet) $ vanish for all $i>N$. Is there any condition on $X$ or $n$ that we can impose so that $H^i(\Omega^\bullet_{W\mathcal{O}_X})$ vanish as well for $i>N$? In other words when does $H^i(T+Fil^n \Omega^\bullet_{W\mathcal{O}_X})$ vanish? Any idea/reference in this direction will be very helpful.