Motivic p-adic tame cohomology

We construct a comparison functor between (${\mathbf{A}}^1$-local) tame motives and ($\bar{\mathrm{\square }}$-local) log-étale motives over a field $k$ of positive characteristic. This generalizes Binda–Park–Østvær’s comparison for the Nisnevich topology. As a consequence, we construct an $E_{\mathrm{\infty }}$-ring spectrum $H\mathbb{Z}\mathrm{/}{p}^m$ representing mod $p^m$ tame motivic cohomology: the existence of this ring spectrum and the usual properties of motives imply some results on tame motivic cohomology, which were conjectured by Hübner–Schmidt.