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

Let $n \geq 1$, $p$ 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 $${^{I}}\mspace{-3mu}E_2^{s,t} \Longrightarrow \pi_{t-s}((L_{T(n+1)}K(E_n))^{hG_n}) \Longleftarrow {^{II}}\mspace{-2mu}E_2^{s,t}$$ with common abutment $\pi_\ast(-)$ of the continuous homotopy fixed points of $L_{T(n+1)}K(E_n)$, where ${^{I}}\mspace{-3mu}E_2^{s,t}$ 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 $${^{I}}\mspace{-3mu}E_2^{\ast,\ast} \cong H^\ast_\mathrm{cts}(G_n, \pi_\ast(L_{T(n+1)}K(E_n))) \cong {^{II}}\mspace{-2mu}E_2^{\ast,\ast}.$$ We isolate two hypotheses, the first of which is true when $(n,p) = (1,2)$, that imply $(L_{T(n+1)}K(E_n))^{hG_n} \simeq L_{T(n+1)}K(L_{K(n)}S^0)$. Also, we show that there is a spectral sequence $$H^s_\mathrm{cts}(G_n, \pi_t(K(E_n) \otimes T(n+1))) \Longrightarrow \pi_{t-s}((K(E_n) \otimes T(n+1))^{hG_n}).$$