Homeomorphism of the Balmer comparison map for all finite groups

Prove that Balmer’s comparison map p: Spc(Cell(G)c) -> Spec(R(G)) is a homeomorphism for every finite group G, where Cell(G) denotes the G-equivariant bootstrap (cell) tt-category and Spc(Cell(G)c) its Balmer spectrum of compact objects, and R(G) the complex representation ring of G.

Background

The authors compute the comparison map p to be a homeomorphism in several cases: for groups whose nontrivial elements have prime order and rationally for arbitrary finite groups. They also note p is always surjective for finite (or compact) G. Establishing this homeomorphism universally would clarify the tt-geometry of Cell(G) and support broader stratification results.

This conjecture sits at the center of understanding the Balmer spectrum of equivariant Kasparov categories for finite groups and would provide a concrete geometric description via Spec(R(G)).