Segal Conjecture for infinite groups

Let $G$ be a group such that the classifying space for proper $G$-actions, $\underline{E}G$, has a finite model. Let $A_{\operatorname{ho}}(G)=\pi_G^0(\underline{E}G)$, let ${\mathbf I}_G=\ker(\epsilon^G)$ be the augmentation ideal, and let $X$ be a finite proper $G$-$CW$-complex. Write $\pi_G^n(X)\widehat{_{\mathbf I_G}}$ for the ${\mathbf I}_G$-adic completion of the $A_{\operatorname{ho}}(G)$-module $\pi_G^n(X)$. Segal Conjecture for infinite groups. There is an isomorphism

$$\pi_s^n(EG\times_GX)\xrightarrow{\cong}\pi_G^n(X)\widehat{_{\mathbf I_G}}.$$

In particular, for all $n\in{\mathbb Z}$,

$$\pi_s^n(BG)\xrightarrow{\cong}\pi_G^n(\underline{E}G)\widehat{_{\mathbf I_G}};$$

and for $n=0$,

$$\pi_s^0(BG)\xrightarrow{\cong}A_{\operatorname{ho}}(G)\widehat{_{\mathbf I_G}}.$$

This is the proposed completion theorem extending the classical Segal conjecture from finite groups to groups with finite models for $\underline{E}G$; the supplied text gives no resolution.