Folklore conjecture on liftings of globally F-split varieties

Let $X$ be a globally $F$-split normal projective variety over an algebraically closed field $k$ of characteristic $p>0$. The variety $X$ is globally $F$-split if its Frobenius morphism admits a splitting. Folklore lifting conjecture. The variety $X$ lifts to a flat projective scheme $\mathcal{X}$ over the ring $W(k)$ of Witt vectors. This conjecture would extend characteristic-zero lifting results beyond the smooth Calabi–Yau case. The source presents the smooth Calabi–Yau conjecture above as a special case of this statement.