Finiteness conjecture for Zariski cohomology of etale cohomology sheaves

Let $X$ be a smooth scheme over one of the reasonable bases ${\Bbb F}_p$, ${\Bbb Z}[1/l]$, or a separably closed field. For all integers $i,j,\nu,n$, consider the Zariski cohomology groups

$$H^i_{\operatorname{Zar}}\left(X,\mathcal H^j_{\operatorname{\acute et}}({\Bbb Z}/l^\nu(n))\right).$$

Finiteness conjecture. These groups are finite. This conjecture is introduced to obtain finite-generation consequences for motivic cohomology when combined with the etale realization conjecture.