Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.

1.If $X$ isFrobenius split(the $p$-th power map $\mathcal{O}_X \to F_* \mathcal{O}_X$ admits n $\mathcal{O}_X$-linear splitting) then the Kodaira vanishing theorem holds for $X$.

The proof uses nothing but Serre vanishing and the projection formula.

2.If the complex $F_* \Omega^\bullet_X$ is quasi-isomorphic to a complex with zero differentials, then the Kodaira-Akizuki-Nakano vanishing theorem holds for $X$.

The proof uses Cartier isomorphism, hypercohomology spectral sequences, Serre vanishing and the projection formula and is similar to that of **1.**

3 (Deligne-Illusie 1987).If $X$ lifts to $W_2(k)$, then the complex $F_* \Omega^\bullet_X$ is quasi-isomorphic to a complex with zero differentials.

4 (Buch-Thomsen-Lauritzen-Mehta 1995).If $X$ is strongly Frobenius split (that is, $0\to B_1\to Z_1\to \Omega^1_X\to 0$ splits, where $Z_i$ and $B_i$ are cocycles/coboundaries in $F_* \Omega^\bullet_X$), then $X$ and $F$ lift to $W_2(k)$ and the Bott vanishing theorem holds for $X$.

My (maybe incorrect) feeling is that strong Frobenius splitting and lifting of the Frobenius to $W_2(k)$ are quite uncommon, Frobenius splitting is a common behavior "on the Fano side" and that lifting of $X$ to usually $W_2(k)$ exists.

Question.Are there examples of Frobenius split varieties for which $F_* \Omega^\bullet_X$ is not quasi-isomorphic to a complex with zero differentials (for example, because the Hodge spectral sequence does not degenerate, see also this question on the Hodge spectral sequence)? If yes (that's my intuition here), does Frobenius splitting imply some weaker property of $F_* \Omega^\bullet_X$ which implies Kodaira vanishing?

**Edit.** Note that Frobenius splitting just states that the complex $F_* \Omega^\bullet_X$ is quasi-isomorphic to a complex whose first differential $C^0 \to C^1$ is zero.