  In this follow-up paper we show that smooth Hodge-proper stacks over
$\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map
between \'etale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic
fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general
conjectures made in our previous work. As a corollary, we get that if a smooth
Artin stack over $K$ has a smooth Hodge-proper model over $\mathcal O_K$, its
$\mathbb Q_p$-\'etale cohomology is a crystalline Galois representation. We
then also establish a truncated version of the above results in more general
setting of smooth $d$-de Rham-proper stacks over $\mathcal O_K$: here we only
require first $d$ de Rham cohomology groups be finitely-generated over
$\mathcal O_K$. As an application, we deduce a certain purity-type statement
for \'etale $\mathbb Q_p$-cohomology of Raynaud generic fiber, as well as
crystallinity of a first several \'etale cohomology groups in the presence of a
Cohen--Macauley model over $\mathcal O_K$ in the schematic setting.
