Syntomification and crystalline local systems

Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{S}\mathrm{p}\mathrm{f}{\mathcal{O}}_K$ where $K\mathrm{/}{\mathbf{Q}}_p$ is a finite extension. We show that reflexive sheaves on the stack ${\mathrm{X}}^{\mathrm{S}\mathrm{y}\mathrm{n}}$ are equivalent to ${\mathbf{Z}}_p$-lattices in crystalline local systems on the rigid generic fiber ${\mathrm{X}}_{\eta }$, and then use this to study the essential image of the étale realization functor on the isogeny category of perfect complexes on ${\mathrm{X}}^{\mathrm{S}\mathrm{y}\mathrm{n}}$. We also show that when $\mathrm{X}\mathrm{/}\mathrm{S}\mathrm{p}\mathrm{f}{\mathcal{O}}_K$ is smooth and proper that $\mathsf{P}\mathsf{e}\mathsf{r}\mathsf{f}({\mathrm{X}}^{\mathrm{S}\mathrm{y}\mathrm{n}})[1\mathrm{/}p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes.