Breuil's lattice conjecture for semistable representations

Let $D=(D,\varphi_D,N,\operatorname{Fil}^\bullet D_K)$ be a weakly admissible filtered $(\varphi,N)$-module such that $\operatorname{gr}^wD_K=0$ for all $w\notin[0,p-1]$. Let $\mathfrak{M}\in\underline{\operatorname{Mod}}_{\mathfrak{S}}(\varphi)^{\leqslant p-1}$ be equipped with a $\boldsymbol{\mathcal{G}}_{K_\infty}$-equivariant embedding $T^*_{\mathfrak{S}}(\mathfrak{M})\hookrightarrow V^*_{\operatorname{st}}(D)$. Breuil's lattice conjecture. The image of this embedding is $\boldsymbol{\mathcal{G}}_K$-stable if and only if $S\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}$ is a strongly divisible $S$-lattice in $S\otimes_{W(k)}D$. Furthermore, if $D$ does not admit a non-zero weakly admissible quotient pure of slope $p-1$, then

$$T^*_{\mathfrak{S}}(\mathfrak{M})=T^*_{\operatorname{st}}(S\otimes_{\varphi,\mathfrak{S}}\mathfrak{M})$$

as $\mathbb{Z}_p$-lattices in $V^*_{\operatorname{st}}(D)$. This is presented as an expected generalization of the paper's method; its resolution is not established in the supplied text.