Generalize Eberhardt’s K-theory proof (Proposition 3.2) to the motives DM setting

Determine whether the proof of Proposition 3.2 in Jens Niklas Eberhardt’s “K-motives, Springer Theory and the Local Langlands Correspondence” (arXiv:2401.13052), which relies on explicit computations in K-theory, can be generalized to the triangulated category of motives DM, and, if so, construct an explicit DM-based proof or method establishing the corresponding result within the motivic framework.

Background

The paper proves an integral version of the motivic Springer correspondence using reduced motives and develops a formalism extending previous rational-coefficient results. In discussing related work, the author notes that Eberhardt claimed a stronger version via K-motives, relying on explicit K-theoretic computations.

The author’s approach replaces these K-theoretic inputs by two corollaries proved in the present paper to obtain the desired equivalence. However, the direct generalization of Eberhardt’s K-theory-based proof to the DM setting is explicitly stated as unclear, leaving open whether a DM-intrinsic proof analogous to Eberhardt’s exists.