Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:

The Hodge-De Rham spectral sequence $E^{a,b}_1=R^af∗Ω^b(X/S)⇒H^{a+b}_{DR}(X/S)$ degenerates in $E_1$.

This is also true when $S=Spec(k)$ with $k$ a perfect field of characteristic $p$, and $X$ has a smooth $W_2(k)$-lifing. Here $W_2(k)$ is the length 2 truncated Witt ring with values in $k$.

My question is when $S= Spec(W_2(k))$, and $X$ has a smooth $W_3(k)$-lifting, does the statement of $E_1$ degeneration still hold (maybe one needs more conditions)? Has anyone worked on this kind of question before?