On Fontaine's conjecture for torsion crystalline local systems

Let $\mathfrak{X}$ be a smooth connected $p$-adic formal scheme. Based on the prismatic description of crystalline local systems, we prove an analogue of Fontaine's conjecture for torsion crystalline local systems on the generic fiber of $\mathfrak{X}$. As an application, we show that the locus of crystalline local systems whose Hodge-Tate weights lie in a fixed interval cuts out a closed subscheme of the universal deformation ring.