Higher rational and Du Bois singularities via motivic oscillation

Let $e$ be a positive integer, and let $Z$ be a locally complete intersection variety of pure dimension over a number field $K$. Define

$$\kappa(Z)=\dim(Z)+\min_{P\in Z(\overline K)}\operatorname{moi}^{\mathrm{aloc}}_{K(P),P}(Z).$$

Here $\operatorname{moi}^{\mathrm{aloc}}_{K(P),P}(Z)$ is the local motivic oscillation index. Higher singularities conjecture. The variety $Z$ has $e$-rational singularities if and only if $\kappa(Z)>e$, and, respectively, has $e$-Du Bois singularities if and only if $\kappa(Z)\geq e$.

This conjecture seeks a characterization of higher rational and higher Du Bois singularities through local motivic oscillation indices. The source explains that the cases $e=0$ are known, while the general positive-$e$ statement remains conjectural.