The weak ordinarity conjecture for projective simple normal crossing varieties

Let $V$ be an $n$-dimensional projective simple normal crossing variety over an algebraically closed field of characteristic zero, and let $A$ be a finitely generated $\mathbb{Z}$-subalgebra over which $V$ has a model. Write $V_\mu$ for the fiber over a closed point $\mu\in\operatorname{Spec}A$. Weak ordinarity conjecture. If the natural Frobenius action on $H^i(V_\mu,\mathcal{O}_{V_\mu})$ is nilpotent for every $i\ge1$ and every closed point in a dense open subset of $\operatorname{Spec}A$, then

$$H^i(V,\mathcal{O}_V)=0$$

for all $i\ge1$. This is the implication needed to relate Frobenius nilpotence of reductions to vanishing of the coherent cohomology of the characteristic-zero variety; no resolution is given in the supplied text.