Achinger–Witaszek–Zdanowicz conjecture on Frobenius-liftable varieties

Let $X$ be a smooth projective variety defined over an algebraically closed field $k$ of characteristic $p>0$, with Frobenius morphism

$$F_X:X\to X.$$

Assume that the pair $(X,F_X)$ admits a flat lifting over $W_2(k)$.

Achinger–Witaszek–Zdanowicz conjecture. There exists a finite étale Galois cover $f:Y\to X$ such that the Albanese morphism

$$Y\to\operatorname{Alb}(Y)$$

admits a structure of a toric fibration. If $X$ is simply connected, then $X$ is a toric variety.

A complete classification of $F$-liftable smooth projective surfaces over $W_2(k)$ is known, but the stated higher-dimensional classification remains open.