The ordinarity conjecture for smooth projective schemes

Let $X$ be a smooth projective scheme defined over a field of characteristic zero. After spreading out $X$, for a prime $p$, write $X_{p=0}$ for its reduction modulo $p$, and let $B\Omega^j_{X_{p=0}}$ denote the sheaf of exact differential $j$-forms on $X_{p=0}$. Ordinarity conjecture. For infinitely many primes $p$, the reduction $X_{p=0}$ is ordinary in the sense of Bloch–Kato, namely

$$H^i(X_{p=0}, B\Omega^j_{X_{p=0}})=0$$

for all $i,j\geq 0$. This conjecture is used to obtain the implication from $k$-Du Bois singularities to $k$-$F$-injectivity in reductions modulo infinitely many primes; its general status is not resolved in the source.