Hodge-Witt symmetry conjecture for liftable varieties

Let $X/k$ be a smooth, proper variety over a perfect field. Assume that $X$ lifts to $W_2$ and that the crystalline cohomology of $X$ is torsion free. For all $i,j$, let $T^{i,j}$ be the dimension of the domino associated to the differential

$$d:H^j(X,W\Omega^i)\to H^j(X,W\Omega^{j+1}).$$

Hodge-Witt symmetry conjecture. The variety $X$ satisfies Hodge-Witt symmetry. In particular, Ekedahl's conditions hold:

$$T^{i,j}=T^{j-2,i+2}.$$

The conjecture proposes Hodge-Witt symmetry under a $W_2$-lifting and torsion-freeness hypothesis; the source presents it as a precise conjectural statement and gives no resolution.