Localization conjecture for strongly self-absorbing equivariant \C^*-algebras

Let $G$ be a compact group and let $D$ be a strongly self-absorbing $G$-$\mathrm{C}^*$-algebra belonging to the $G$-equivariant bootstrap class $\mathcal{B}_G$. A separable $G$-$\mathrm{C}^*$-algebra $B$ is compact if it is a compact object of the $G$-equivariant Kasparov category $KK^G$, meaning that for every countable family $\{C_n\}_n$ of separable $G$-$\mathrm{C}^*$-algebras, the canonical map

$$\bigoplus_n KK^G(B,C_n)\cong KK^G\left(B,\bigoplus_n C_n\right)$$

is an isomorphism. The algebra $D$ satisfies the localization condition with respect to $B$ if there exist $R(G)$-module isomorphisms

$$h_i^B:KK_i^G(B,\mathbb{C})_{S_D}\xrightarrow{\cong}KK_i^G(B,D),\qquad i=0,1,$$

that identify the natural localization maps with the maps induced by the unit $\nu:\mathbb{C}\to D$.

Localization conjecture. $D$ satisfies the localization condition with respect to every separable compact $G$-$\mathrm{C}^*$-algebra $B$.

This conjecture proposes a uniform localization description of equivariant Kasparov theory for strongly self-absorbing $G$-$\mathrm{C}^*$-algebras in the equivariant bootstrap class. Its resolution is not specified in the source.