Several homotopy fixed point spectral sequences in telescopically localized algebraic $K$-theory

Let $n\ge 1$, $p$ a prime, and $T(n)$ any representative of the Bousfield class of the telescope $v_n^{-1}F(n)$ of a finite type $n$ complex. Also, let $E_n$ be the Lubin-Tate spectrum, $K(E_n)$ its algebraic $K$-theory spectrum, and $G_n$ the extended Morava stabilizer group, a profinite group. Motivated by an Ausoni-Rognes conjecture, we show that there are two spectral sequences $${}^I\text{mspace}-3mu{E}_2^{s,t}⟹{\pi }_{t-s}((L_{T(n+1)}K(E_n){)}^{h{G}_n})⟸{}^{II}\text{mspace}-2mu{E}_2^{s,t}$$ with common abutment ${\pi }_{\ast }(-)$ of the continuous homotopy fixed points of $L_{T(n+1)}K(E_n)$, where ${}^I\text{mspace}-3mu{E}_2^{s,t}$ is continuous cohomology with coefficients in a certain tower of discrete $G_n$-modules. If the tower satisfies the Mittag-Leffler condition, then there are continuous cochain cohomology groups $${}^I\text{mspace}-3mu{E}_2^{\ast ,\ast }\cong H_{\mathrm{c}\mathrm{t}\mathrm{s}}^{\ast }(G_n,{\pi }_{\ast }(L_{T(n+1)}K(E_n)))\cong {}^{II}\text{mspace}-2mu{E}_2^{\ast ,\ast }\mathrm{.}$$ We isolate two hypotheses, the first of which is true when $(n,p)=(1,2)$, that imply $(L_{T(n+1)}K(E_n){)}^{h{G}_n}\simeq L_{T(n+1)}K(L_{K(n)}{S}^0)$. Also, we show that there is a spectral sequence $$H_{\mathrm{c}\mathrm{t}\mathrm{s}}^s(G_n,{\pi }_t(K(E_n)\otimes T(n+1)))⟹{\pi }_{t-s}((K(E_n)\otimes T(n+1){)}^{h{G}_n})\mathrm{.}$$