Nucleus equals support of the singularity category for finite groups

Determine whether, for a finite group G and a field k, the nucleus Y_G equals the support of the singularity category sg(C^*(BG)), where C^*(BG) denotes the differential graded algebra of cochains on the classifying space BG and sg(-) denotes the singularity category in the sense of Greenlees and Stevenson, to extend the correspondence established here for compact Lie groups.

Background

The paper develops a notion of support for singularity categories of ring spectra and shows that for compact Lie groups in characteristic not dividing the Weyl group order, this support coincides precisely with the nucleus, which in turn matches the homogeneous singular locus of H^*(BG).

The authors reference a conjecture from Benson and Greenlees (Conjecture 7.5 in their preprint) asserting the same correspondence for finite groups. While the present work proves the correspondence for compact Lie groups under non-modular conditions, the finite-group case is referred to as a conjecture and remains an explicit open problem in that context.