A construction of some objects in many base cases of an Ausoni-Rognes conjecture

Let $p$ be a prime, $n geq 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, $mathbb{G}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and Rognes conjectured that (a) the unit map $i_n: L_{K(n)}(S^0) to E_n$ from the $K(n)$-local sphere to the Lubin-Tate spectrum induces a map [K(L_{K(n)}(S^0)) wedge v_{n+1}^{-1}V_n to (K(E_n))^{hmathbb{G}_n} wedge v_{n+1}^{-1}V_n] that is a weak equivalence, where (b) since $mathbb{G}_n$ is profinite, $(K(E_n))^{hmathbb{G}_n}$ denotes a continuous homotopy fixed point spectrum, and (c) $pi_ast(-)$ of the target of the above map is the abutment of a homotopy fixed point spectral sequence. For $n = 1$, $p geq 5$, and $V_1 = V(1)$, we give a way to realize the above map and (c), by proving that $i_1$ induces a map [K(L_{K(1)}(S^0)) wedge v_{2}^{-1}V_1 to (K(E_1) wedge v_{2}^{-1}V_1)^{hmathbb{G}_1},] where the target of this map is a continuous homotopy fixed point spectrum, with an associated homotopy fixed point spectral sequence. Also, we prove that there is an equivalence [(K(E_1) wedge v_{2}^{-1}V_1)^{hmathbb{G}_1} simeq (K(E_1))^{widetilde{h}mathbb{G}_1} wedge v_2^{-1}V_1,] where $(K(E_1))^{widetilde{h}mathbb{G}_1}$ is the homotopy fixed points with $mathbb{G}_1$ regarded as a discrete group.