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Nonabelian Hodge theory in positive characterstic via exponential twisting

Let $k$ be a perfect field of odd characteristic and $X$ a smooth algebraic variety over $k$ which is $W_2$-liftable. We show that the exponent twisiting of the classical Cartier descent gives an equivalence of categories between the category of nilpotent Higgs sheaves of exponent $\leq p-1$ over $X/k$ and the category of nilpotent flat sheaves of exponent $\leq p-1$ over $X/k$, and it is equivalent up to sign to the inverse Cartier and Cartier transforms for these nilpotent objects constructed in the nonabelian Hodge theory in positive characteristic by Ogus-Vologodsky. In view of the crucial role that Deligne-Illusie's lemma has ever played in their algebraic proof of $E_1$ degeneration and Kodaira vanishing theorem in abelian Hodge theory, it may not be overly surprising that again this lemma plays a significant role via the concept of Higgs-de Rham flow in establishing $p$-adic Simpson correspondence in nonabelian Hodge theory and Langer's algebraic proof of Bogomolov inequality for semistable Higgs bundles and Miyaoka-Yau inequality.