The ordinarity conjecture for smooth projective varieties

Let $X$ be a smooth, irreducible $n$-dimensional projective variety defined over an algebraically closed field $k$ of characteristic zero. If $X_A$ is a model of $X$ defined over a subring $A$ of $k$, finitely generated over $\mathbf{Z}$, and $X_s$ denotes its fiber at a closed point $s\in\operatorname{Spec} A$, then consider the Frobenius action on $H^n(X_s,\mathcal{O}_{X_s})$. The ordinarity conjecture. There is a dense set of closed points $S\subseteq\operatorname{Spec} A$ such that the Frobenius action on

$$H^n(X_s,\mathcal{O}_{X_s})$$

is bijective for every $s\in S$. This was proposed in the cited work with V. Srinivas and is a characteristic-positive ordinarity condition expected to occur densely among reductions; the source presents it as conjectural and proves that the multiplier–test ideal comparison conjecture implies it.