A particular case of Lurie and Hopkins' ambidexterity theory is that if $G$ is a finite group acting on a $K(n)$-local spectrum $X$ then the norm map
$$ X\_{hG} \to X^{hG} $$
is a $K(n)$-local equivalence. If we now replace $G$ with a compact lie group and $X$ with a genuine equivariant **free** $G$-spectrum (not necessarily $K(n)$-local), then the Adams equivalence asserts that the twisted norm map 
$$ X\_{G} \to (\Sigma^{-ad(G)} X)^G $$
is an equivalence, where $ad(G)$ is the adjoint representation of $G$. One might then be led to ponder the possibility that these two phenomena are somehow part of the same thing. For example, one might consider the following (possibly overly optimistic) conjecture:


**Conjecture:**
Let $G$ be a compact lie group and $X$ a genuine equivariant $K(n)$-local $G$-spectrum (not necessarily free). Then the twisted norm map
$$ X\_{G} \to (\Sigma^{-ad(G)} X)^G $$
is a $K(n)$-local equivalence.


Is this conjecture known to be true or false? I've had a hard time finding references which talk about genuine equivariance and ambidexterity at the same time, so any literature pointers which might be relevant to the above will be most appreciated. 



Genuine equivariant ambidexterity
This is to address Yonatan's question in the comments. Let $G$ be a finite group. To every (genuine) $G$-spectrum $E$, you can associate its (genuine) fixed point spectrum $E^{G}$. This earns its name by virtue of the following compatibility between stable and unstable homotopy theory:
$\Omega^{\infty}( E^G ) = (\Omega^{\infty} E)^{G}$.


Now you could ask, could we have some categorically dual gadget
$E \mapsto E\_{G}$, which satisfied the dual condition
$\Sigma^{\infty}\_{+} (X/G) = (\Sigma^{\infty}\_{+} X)\_{G}$ for $X$ a $G$-space? The answer is no, because the construction on the left hand side
$X \mapsto \Sigma^{\infty}\_{+} (X/G)$ is functorial with respect to maps of $G$-spaces, but not with respect to stable maps of $G$-spaces.


For example, any correspondence of $G$-spaces $X \leftarrow M \rightarrow Y$ where $M$ is a finite covering space of $X$ determines a map
of $G$-spectra from $\Sigma^{\infty}\_{+}(X)$ to $\Sigma^{\infty}\_{+}(Y)$.
But there's no functorial procedure to extract from $M$ a map of spectra
$\Sigma^{\infty}\_{+} (X/G) \rightarrow \Sigma^{\infty}\_{+}(Y/G)$.
You might say: why not use the correspondence of nonequivariant spaces given by $X/G \leftarrow M/G \rightarrow Y/G$? This does not respect composition of correspondences (observe what happens when you compose the correspondence $\ast \leftarrow G \rightarrow \ast$ with itself).


