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Let $H$ be a closed subgroup of the Morava stabilizer group $\mathbb G_n$. [Devinatz-Hopkins, Prop. 6.7] identifies the $K(n)$-local $E_n$-Adams spectral sequence for $E_n^{hH}$ as the homotopy fixed point spectral sequence $$H^s_c\left(H;(E_n)_t\right)\Rightarrow\pi_{t-s}E_n^{hH}.$$ On the other hand, we also have the Adams-Novikov spectral sequence $$\operatorname{Ext}^{s,t}_{BP_*BP}\left(BP_*,BP_*(E_n^{hH})\right)\Rightarrow\pi_{t-s}E_n^{hH}.$$

Question. Are these two spectral sequences isomorphic?

When I look into the special case at $n=1$, $p=2$: $$H^s\left(C_2;(E_1)_t\right)\Rightarrow\pi_{t-s}E_1^{hC_2},$$ the pattern seems exactly the same as the ANSS for $KO$. I think there is a map of spectral sequences from ANSS to HFPSS induced by $BP\to E_n$, but I cannot see if the map of spectral sequences is an isomorphism.

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    $\begingroup$ The first spectral sequence can also be identified with the $K(n)$-local $E_n^{hH}$-Adams spectral sequence by ams.org/journals/tran/2005-357-01/S0002-9947-04-03394-X/…. It looks like the claimed isomoprhism might follow from $K(n)$-localization, and then some version of Morava's change of rings theorem. $\endgroup$
    – Drew Heard
    Commented Oct 4 at 10:41
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    $\begingroup$ BTW: For finite $G$, I believe the result you want follows from Corollary 4.2 of arxiv.org/pdf/2407.01507 $\endgroup$
    – Drew Heard
    Commented Oct 4 at 11:18

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