Schwede–Smith conjecture on globally F-regular type varieties

Let $X$ be a variety over $\a0\mathbb{C}$ that is of globally $F$-regular type, meaning that it has a model over a finitely generated $\a0\mathbb{Z}$-algebra whose reductions are globally $F$-regular for a suitable dense set of closed points.

Schwede–Smith conjecture. $X$ is of Fano type.

This conjecture relates characteristic-$p$ positivity properties to the birational-geometric notion of Fano type. The source states that it is proved for surfaces, while the general case is not resolved there.