The spectrum conjecture for finite-group equivariant KK-theory

Let $G$ be a finite group, let $\mathcal K^G=\langle 1\!\!1\rangle\subset\mathsf{KK}^G$ be the thick triangulated subcategory generated by the tensor unit, and let $R(G)$ be the complex representation ring of $G$. Consider the natural map

$$\rho_{\mathcal K^G}:\mathop{\mathrm{Spc}}(\mathcal K^G)\longrightarrow\mathop{\mathrm{Spec}}(R(G)).$$

Spectrum conjecture. For every finite group $G$, the natural map

$$\rho_{\mathcal K^G}:\mathop{\mathrm{Spc}}(\mathcal K^G)\longrightarrow\mathop{\mathrm{Spec}}(R(G))$$

is a homeomorphism.

This predicts that the tensor-triangular spectrum of the compact equivariant KK-category generated by the tensor unit is completely described by the prime spectrum of the representation ring. The surrounding discussion presents it as a natural generalization of the identification of the Bootstrap category's spectrum with $\operatorname{Spec}(\mathbb Z)$; no resolution status is supplied here.