The étale and Zariski motivic complex comparison conjectures for p-adic étale Tate twists

Let $X$ be in the setting of the paper, let $n$ and $r$ be the relevant integers, and let ${\Bbb Z}(n)_X^{{\mathrm{\acute{e}t}}}$ and ${\Bbb Z}(n)_X^{{\mathrm{Zar}}}$ denote the étale and Zariski sheafifications of Bloch's cycle complex. Let $\varepsilon$ be the natural continuous map of sites

$$X_{{\mathrm{\acute{e}t}}}\longrightarrow X_{{\mathrm{Zar}}}.$$

Comparison conjecture. (1) There is an isomorphism in $D^b(X_{{\mathrm{\acute{e}t}}},{\Bbb Z}/p^r{\Bbb Z})$

$${\Bbb Z}(n)_X^{{\mathrm{\acute{e}t}}}\otimes^{{\Bbb L}}{\Bbb Z}/p^r{\Bbb Z}\simeq {\frak T}_r(n)_X.$$

(2) The isomorphism in (1) induces an isomorphism in $D^b(X_{{\mathrm{Zar}}},{\Bbb Z}/p^r{\Bbb Z})$

$${\Bbb Z}(n)_X^{{\mathrm{Zar}}}\otimes^{{\Bbb L}}{\Bbb Z}/p^r{\Bbb Z}\simeq \tau_{\leq n}R\varepsilon_*{\frak T}_r(n)_X.$$

These expectations concern the relationship between the newly constructed semistable-family object ${\frak T}_r(n)_X$ and motivic complexes. The paper presents the motivic complexes as strong candidates and derives these statements as hoped-for consequences, but no resolution is supplied here.