Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the homotopy fixed points of the action of $C_2$ on the motivic algebraic $K$-theory spectrum $KGL$ give the $\eta$-completion of the Hermitian $K$-theory spectrum $KQ$, where $\eta$ is the (Morel's) algebraic version of the Hopf map. My question is: is it known whether $KQ$ is is $\eta$-complete in this context, i.e., that the $\eta$-completion operation does not really change anything?

Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the homotopy fixed points of the action of $C_2$ on the motivic algebraic $K$-theory spectrum $KGL$ give the $\eta$-completion of the Hermitian $K$-theory spectrum $KQ$, where $\eta$ is the (Morel's) algebraic version of the Hopf map. My question is: is it known whether $KQ$ is $\eta$-complete in this context, i.e., whether the $\eta$-completion operation does not really change anything?