Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?

Question

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?

Answer 1

If I understand the question correctly, you're asking if there are base fields $k$ of characteristic not $2$ and such that $k[\sqrt{-1}]$ has a finite $\mathbb{Z}/2$-cohomological dimension (the assumptions used in the above mentioned paper) for which the map $KQ \to KGL^{hC_2}$ is not an equivalence. Note that by the Tate fiber square (mentioned also in that paper), this is the same as asking whether $KW \to KGL^{tC_2}$ is an equivalence, a necessary condition for which is that the map $W(k) \to \pi_0(K(k)^{tC_2})$ is an isomorphism of abelian groups, where $W(k)$ is the Witt group of $k$ and $K(k)$ is its $K$-theory spectrum. As shown in Theorem 16 of this paper, the last map coincides in this context with the completion map $$ (*) \quad W(k) \to \lim_n W(k)/I^n $$ with respect to the augmentation ideal $I \subseteq W(k)$. An example of a field satisfying the above conditions and for which this completion map is not an isomorphism is the field $\mathbb{R}$ of real numbers, where $(*)$ is the map $\mathbb{Z} \to \lim_n\mathbb{Z}/2^n$.