Reductions of non-lc ideals and non $F$-pure ideals assuming weak ordinarity

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 $\mathfrak{a}$.
Comment: 11 pages; v2: improved and simplified exposition