# Does the pro-étale local system defined over a p-adic period domains interpolate crystalline representations?

There is a Grothendieck-Messing period morphism of rigid-analytic spaces $\pi: \mathcal{M}_\eta^{rig}\to \mathcal{Fl}$ going from the generic fiber of an EL-type Rapoport-Zinks to a flag variety. The image of this morphism $\mathcal{Fl}^{adm}$ is an open adic space called the admissible locus of the p-adic period domain. In this setting one can take the universal $p$-divisible group with structure and take it's corresponding Tate-module. The data of the Tate-Module allows one to glue a $\mathbb{Q}_p$-local system on $\mathcal{Fl}^{adm}$ whose pullback to $K$-points for $K/\mathbb{Q}_p^{un}$ a finite extensions are crystalline representations. This map has been greatly generalized to a period morphism going from a moduli space of mixed characteristic shtukas $\pi:Sht_{(G,b,\mu)}\to Gr^{Bdr}_{\leq \mu}$ to a so-called $B_{dR}$ Grasmannian bounded by the paw of the shtuka $\mu: \mathbb{G}_m\to G$. There is again a b-admissible open locus together with a pro-etale local system on this Grasmanian. I was wondering if in this generality it is also true that the $K$ pullbacks of this map would be crystalline representations, and if there is a reference available for this fact.