Beilinson–Lichtenbaum conjectures for motivic complexes on arithmetic schemes

Let ${\mathscr X}$ be the arithmetic scheme in the paper, with Zariski and small étale sites ${\mathscr X}_{{\mathrm{Zar}}}$ and ${\mathscr X}_{{\mathrm{\acute{e}t}}}$, and let $\epsilon:{\mathscr X}_{{\mathrm{\acute{e}t}}}\to {\mathscr X}_{{\mathrm{Zar}}}$ be the natural map of sites. Let ${\mathbb Z}(2)_{{\mathrm{Zar}}}$ and ${\mathbb Z}(2)_{{\mathrm{\acute{e}t}}}$ be the Zariski motivic complex and its étale sheafification. Beilinson–Lichtenbaum conjectures. The following statements hold:

$${\mathbb Z}(2)_{{\mathrm{Zar}}}\simeq\tau_{\leq 2}R\epsilon_*{\mathbb Z}(2)_{{\mathrm{\acute{e}t}}}\quad\text{in }D({\mathscr X}_{{\mathrm{Zar}}}).$$

$$R^3\epsilon_*{\mathbb Z}(2)_{{\mathrm{\acute{e}t}}}=0.$$

$$({\mathbb Z}(2)_{{\mathrm{\acute{e}t}}})|_{{\mathscr X}[p^{-1}]}\otimes^{\mathbb L}{\mathbb Z}/p^r{\mathbb Z}\simeq\mu_{p^r}^{\otimes 2}.$$

The source says these conjectures hold when ${\mathscr X}$ is smooth over $S$, by results of Geisser and the Merkur'ev–Suslin theorem, so this candidate is treated as solved in that case rather than as an unresolved conjecture.