Let X be a variety with lci singularities, and let k-Du Bois and (k-1)-rational denote the higher Du Bois and higher rational singularity conditions. Higher Du Bois implies lower rational conjecture. If X has lci singularities and X is k-Du Bois, then X is (k-1)-rational. This conjecture was made in the first version of the paper and reflects the close relationship between higher rational and higher Du Bois singularities, particularly for isolated complete intersections. The supplied text gives no resolution status.

The converse higher Du Bois–rational singularities conjecture

The conjecture asserts that for a variety with lci singularities, $k$ -Du Bois singularities imply $(k-1)$ -rational singularities. It was formulated in the first version of the relevant paper; the scan gives no proposer or precise formulation date.

Known results

Isolated lci singularities: established by the paper’s authors; year not specified in the supplied sources.
Hypersurface singularities: established through work of Mustaţă and Popa and related results; year not specified in the supplied sources.

General proof claim (2022; exact date unspecified)

The paper Higher Du Bois and higher rational singularities reports that Chen, Dirks, and Mustaţă proved the conjecture in general. 2023 lecture slides state the same implication, but the supplied record contains no proof details or independent verification; this is therefore an unverified claimed solution.

Current status (as of September 2026): A general proof has been reported by Chen, Dirks, and Mustaţă, but it has not been independently verified in the supplied record.

The converse higher Du Bois–rational singularities conjecture

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