Dense F-split type for complex Abelian varieties and log Calabi–Yau varieties

Let $X$ be a complex Abelian variety, or more generally a log Calabi–Yau variety. A complex variety has dense $F$-split type if, for a model over a finitely generated $\bZ$-algebra, its fibers are $F$-split at a dense set of closed points. Dense F-split type conjecture. $X$ has dense $F$-split type. This predicts that these characteristic-zero varieties admit many globally $F$-split reductions, linking Frobenius splitting in positive characteristic with the geometry of varieties having trivial or suitably controlled canonical class.