The Segal conjecture describes the Spanier-Whitehead dual $D \Sigma^\infty\_+ BG$ for certain $G$. Is there a similar description of $D\Sigma^\infty\_+ K(G,n)$ when $n \geq 2$ when $G$ is finite (and abelian)?


Notes:


* I'd be happy to understand the case of cyclic groups $G = C\_p$.
* $K(G,n)$ can be modeled by an abelian topological group, but I'm not sure it falls under the umbrella of other known generalizations of the Segal conjecture, although when $G = \mathbb Z$ and $n=2$ there is a known decomposition (see [Ravenel](https://web.math.rochester.edu/people/faculty/doug/mypapers/segal.pdf)). For $G = \mathbb Z^n$ and $n=2$ there is also [this](https://arxiv.org/abs/0803.2745).
* Let me recall that the Segal conjecture (proved by Carlsson) says that when $G$ is finite, the Spanier-Whitehead dual $D\Sigma^\infty\_+ BG$ is a certain completion of $\vee\_{(H) \subseteq G} \Sigma^\infty\_+ BW\_G(H)$ where $(H) \subseteq G$ ranges over conjugacy classes of subgroups and $W\_G(H) = N\_G(H) / H$ is the Weyl group of $H$ in $G$. In particular, when $G = C\_p$ it says that 


$$D\Sigma^\infty\_+ BC\_p = \mathbb S \vee(\Sigma^\infty\_+ BC\_p )^{\wedge}\_p$$


where $\mathbb S$ is the sphere spectrum (corresponding to the subgroup $C\_p \subseteq C\_p$; the other term corresponds to the trivial subgroup $0 \subseteq C\_p$) and $(-)^\wedge\_p$ is $p$-completion.
* Lin [showed](https://doi.org/10.1090/S0002-9939-1976-0402738-5) that $D H G = 0$ when $G$ is a finite abelian group, where $H$ indicates taking Eilenberg-MacLane spectra. Since $HG = \varinjlim\_n \Sigma^{\infty-n} K(G,n)$, we have $0 = DHG = \varprojlim\_n \Sigma^n D\Sigma^\infty K(G,n)$, and from the Milnor exact sequence we conclude that $\varprojlim\_n \pi\_{\ast-n} D\Sigma^\infty K(G,n) = \varprojlim^1\_n \pi\_{\ast-n} D \Sigma^\infty K(G,n) = 0$. But I'm not sure how much information that is, really.
* If we work in the $K(h)$-local or the $T(h)$-local category then by ambidexterity we have $F(\Sigma^\infty\_+ K(G,n), L\mathbb S) = L \Sigma^\infty\_+ K(G,n)$ where $L$ is the relevant localization. But it seems that the relevant limit does not commute with localization here.



Is there a "higher Segal conjecture"?
In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(\mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea: view $K(A,n+1)$ as the bar construction on $K(A,n)$.)


(No time right now to write more ... but maybe this is enough.)


