It is known in classical stable homotopy theory that there is an equivalence
$$ L_{K(1)}S\simeq KU^{h\mathbb{Z}_p^\times} $$ which gives an especially convenient way to compute the K(1)-local sphere. Motivically, we have analogues of $KU$ and $K(1)$. These are given, respectively, by the motivic spectrum representing algebraic $K$-theory $KGL$ and the algebraic Morava $K$-theory $K(1)$ as constructed by Borghesi. I am under the impression that the Adams operations also act $KGL$ via $E_\infty$-maps, and so it makes sense to write $KGL^{h\mathbb{Z}_p^\times}$. Is it known if there is analogous equivalence
$$ L_{K(1)}S^{0,0}\simeq KGL^{h\mathbb{Z}_p^\times}? $$
A more down to earth version of this question might be the following. Classically, we also have (in the 2-complete category) the Adams-Baird fibre sequence
$$ S_{K(1)}\to KO\to KO $$ where the map $KO\to KO$ is $\psi^3-1$. Motivically, the analog of $KO$ is the motivic spectrum $KQ$ representing Hermitian $K$-theory. Is it known whether or not the corresponding fibre sequence exists motivically?