Fontaine's conjecture for torsion crystalline representations

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field $k$, and let $G_K=\operatorname{Gal}(\overline{K}/K)$. Fix a positive integer $r$, and let $T$ be a finite free $\mathbf{Z}_p$-representation of $G_K$. For each $n\geq 1$, suppose that $T/p^nT$ is torsion crystalline with Hodge--Tate weights in $[0,r]$, meaning that there exist $G_K$-stable $\mathbf{Z}_p$-lattices $T_2^{(n)}\subset T_1^{(n)}$ in a crystalline representation with Hodge--Tate weights in $[0,r]$ such that

$$T/p^nT\cong T_1^{(n)}/T_2^{(n)}$$

as $\mathbf{Z}_p[G_K]$-modules. Fontaine's conjecture. Then $T[p^{-1}]$ is crystalline with Hodge--Tate weights in $[0,r]$. This predicts that the crystalline condition is detected on all torsion reductions and is the torsion analogue of Fontaine's prediction concerning crystalline deformation rings. The statement is proved by Liu.