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?

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