Reduction of maximal non-lc ideals to non-$F$-pure ideals

Let $X$ be a normal variety over a field $k$ of characteristic zero, let $b\Deltab$ be an effective $b\mathbb{Q}b$-divisor such that $K_X + \Delta$ is $b\mathbb{Q}b$-Cartier, and let $\mathfrak{a}$ be an ideal sheaf in $\mathcal{O}_X$. Given any model of $(X, \Delta, \mathfrak{a})$ over a finitely generated $\mathbb{Z}$-subalgebra $A$ of $k$, there exists a dense set of closed points $S$ of $\operatorname{Spec} A$ such that

$$\sigma(X_s, \Delta_s, \mathfrak{a}_s^\lambda)=\mathcal{J}'(X, \Delta, \mathfrak{a}^\lambda)_s$$

for all $\lambda \geq 0$ and all $s \in S$.

Maximal non-lc reduction conjecture. The reduction of the maximal non-lc ideal filtration coincides with the non-$F$-pure ideal filtration on a dense set of closed fibers, as asserted above.

The conjecture is presented as the characteristic-zero/positive-characteristic analogue for maximal non-lc and non-$F$-pure ideals, and the paper proves that it follows from the weak ordinarity conjecture. The parser supplies no resolution evidence for this statement.