## A p-divisible puzzle

Let $\mathbf{C}_p$ be a completed algebraic closure of $\mathbf{Q}_p$, and let $G/ \mathcal{O}_{\mathbf{C}_p}$ be a p-divisible group. The Tate module of $G$ sits in a Hodge-Tate sequence

${0\rightarrow\mathrm{Lie}G\otimes_{\mathbf{Z}_{p}}\mathbf{Q}_{p}(1)\rightarrow T_{p}G\otimes_{\mathbf{Z}_{p}}\mathbf{C}_{p}\rightarrow\mathrm{Lie}(G^{\ast})^{\ast}\otimes_{\mathbf{Z}_{p}}\mathbf{Q}_{p}\rightarrow0}$.

According to a recent theorem of Scholze-Weinstein, the map $G \rightsquigarrow (T_pG, \mathrm{Lie}(G))$ induces an equivalence of categories from the category of p-divisible groups over $\mathcal{O}_{\mathbf{C}_p}$ to the category of pairs $(\Lambda,W)$ where $\Lambda$ is a free finite rank $\mathbf{Z}_p$-module and $W \subset \Lambda \otimes_{\mathbf{Z}_p}\mathbf{C}_p(-1)$ is a $\mathbf{C}_p$-sub-vector-space.

So here’s the puzzle. Let $A$ be an abelian variety over $\mathbf{C}_p$; this has an analogous Hodge-Tate sequence, and the pair $(T_pA, \mathrm{Lie}(A))$ gives rise to a p-divisible group $G/ \mathcal{O}_{\mathbf{C}_p}$.  What is the relationship between $G$ and $A$ when $A$ has bad reduction?