Balmer comparison-map conjecture for equivariant cell categories

Let $G$ be a finite group, let $\Cell{G}$ be the category of $G$-cell algebras, and let $\Spc(\Cell{G}^c)$ be its tensor-triangular spectrum. The tensor unit is the trivial $G$-C*-algebra, with endomorphism ring $\operatorname{End}(\unit)=\Rep(G)$, and there is a canonical comparison map

$$\rho\colon \Spc(\Cell{G}^c)\longrightarrow \operatorname{Spec}(\Rep(G)).$$

Balmer's comparison-map conjecture. For every finite group $G$, the comparison map is a homeomorphism

$$\Spc(\Cell{G}^c)\cong \operatorname{Spec}(\Rep(G)).$$

The map is known to be surjective for finite groups, and the paper proves that it is a homeomorphism when every nontrivial element of $G$ has prime order. Its being a homeomorphism for arbitrary finite groups remains open and is an obstacle to proving general countable stratification.