  Assume $X$ is a variety over $\mathbb{C}$, $A \subseteq \mathbb{C}$ is a
finitely generated $\mathbb{Z}$-algebra and $X_A$ a model of $X$ (i.e. $X_A
\times_A \mathbb{C} \cong X$). Assuming the weak ordinarity conjecture we show
that there is a dense set $S \subseteq \text{Spec } A$ such that for every
closed point $s$ of $S$ the reduction of the maximal non-lc ideal filtration
$\mathcal{J}'(X, \Delta, \mathfrak{a}^\lambda)$ coincides with the non-$F$-pure
ideal filtration $\sigma(X_s, \Delta_s, \mathfrak{a}_s^\lambda)$ provided that
$(X, \Delta)$ is klt or if $(X, \Delta)$ is log canonical, $\mathfrak{a}$ is
principal and the non-klt locus is contained in $\mathfrak{a}$.
