  Following ideas of Kedlaya-Liu, we are going to consider extending our
previous work to the context of more general adic spaces, which will be
corresponding deformation of the relative $p$-adic Hodge structure over more
general adic spaces. This means that the deformation could be also realized by
an adic spaces (perfectoid, preperfectoid, relatively perfectoid and so on).
Parts of the whole project here actually are inspired by the corresponding
Drinfeld's lemma for diamonds after Scholze, as well as the work from
Carter-Kedlaya-Z\'abr\'adi which is aimed at studying the representation theory
of products of \'etale fundamental groups. Moreover, we gain motivations from
noncommutative analytic geometries and noncommutative Tamagawa number
conjectures.
