Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost sense (i.e. considering the category of extensions in the sense of "almost mathematics"). I want to know whether this is true of other rigid invariants. The question I'm interested in specifically is this: consider the categories $\operatorname{Mod}^a(R), \operatorname{Mod}^a(R')$ of almost modules over $R, R'$. Is it true that (making suitable definitions) the periodic cyclic homology of these two categories over $\mathbb{Z}$ is isomorphic, or related by a spectral sequence?

More generally, I'm looking for interesting tilting statements visible on the level of comparing the categories $\operatorname{Mod}^a(R)$ and $\operatorname{Mod}^a(R')$ (possibly with monoidal structure).