Perfectoid splitting conjecture for smooth proper Calabi–Yau families

Let $S$ be a scheme smooth and dominant over $\operatorname{Spec} \mathbb{Z}$, and let $Y_S\to S$ be a smooth proper morphism whose fibers are Calabi–Yau varieties. For a closed point $\mathfrak p\in S$, set

$$Y_{S,\mathfrak p}:=Y_S\times_S\operatorname{Spec}\mathcal{O}_{S,\mathfrak p}.$$

A quasi-compact separated scheme $X$ is perfectoid split if there exists an affine morphism $\pi:Y\to X$ such that the $p$-adic formal completion $\widehat{Y}$ is a perfectoid formal scheme and the natural map $\mathcal{O}_X\to\pi_*\mathcal{O}_Y$ is ind-split, meaning a filtered colimit of split morphisms in $\operatorname{QCoh}(X)$. Perfectoid splitting conjecture. The localization $Y_{S,\mathfrak p}$ is perfectoid split for a Zariski-dense set of closed points $\mathfrak p\in S$. This is presented as a mixed-characteristic counterpart of the weak ordinarity conjecture and is motivated by perfectoid geometry and its applications to Kodaira-type vanishing.