Segal's conjecture describes the double SpanierWhitehead $ D Sigma ^ infty_ + BG $ it's safe $ G $. Is there a similar description of $ D Sigma ^ infty_ + K (G, n) $ when $ n geq 2 $ when $ G $ is finished (and abelian)?
Remarks:

I would 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 that this falls under the guise of other generalizations known to the Segal conjecture, although $ G = mathbb Z $ and $ n = $ 2 there is a known decomposition (see Ravenel). For $ G = mathbb Z ^ n $ and $ n = $ 2 there is that too.

Let me remind you that Segal's conjecture (proven by Carlsson) says that when $ G $ is finished, the double SpanierWhitehead $ D Sigma ^ infty_ + BG $ is a certain completion of $ vee _ {(H) subseteq G} Sigma ^ infty_ + BW_G (H) $ or $ (H) subseteq G $ ranges on subgroup conjugation classes and $ W_G (H) = N_G (H) / H $ is the Weyl group of $ H $ in $ G $. In particular, when $ G = C_p $ he says that
$$ D Sigma ^ infty_ + BC_p = mathbb S vee ( Sigma ^ infty_ + BC_p) ^ { coin} _p $$
or $ mathbb S $ is the spectrum of the sphere (corresponding to the subgroup $ C_p subseteq C_p $; the other term is the trivial subgroup $ 0 subseteq C_p $) and $ () ^ wedge_p $ is $ p $completion.