Achinger–Zdanowicz conjecture on Witt-vector liftings of Frobenius-split varieties

Let $X$ be a smooth projective variety defined over an algebraically closed field $k$ of characteristic $p>0$. Assume that $X$ is globally Frobenius-split and has trivial canonical class.

Achinger–Zdanowicz conjecture. There is a flat deformation of $X$ over the Witt vectors $W(k)$, not only over $W_2(k)$.

This conjecture concerns extending liftings of varieties with strong Frobenius-splitting properties from truncated Witt vectors to the full Witt vectors. It is known for surfaces and for finite étale quotients of Abelian varieties, while the general case remains open.