Log $p$-divisible groups and semi-stable representations

Let $\mathscr{O}_K$ be a henselian DVR with field of fractions $K$ and residue field of characteristic $p>0$. Let $S$ denote $\mathop{\mathrm{Spec}} \mathscr{O}_K$ endowed with the canonical log structure. We show that the generic fiber functor $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\to \mathbf{BT}^{\mathrm{st}}_K$ between the category of dual representable log $p$-divisible groups over $S$ and the category of $p$-divisible groups with semistable reduction over $K$ is an equivalence. If $\mathscr{O}_K$ is further complete with perfect residue field and of mixed characteristic, we show that $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}$ is also equivalent to the category of semistable Galois $\mathbb{Z}_p$-representations with Hodge-Tate weights in $\{0,1\}$. Finally, we show that the above equivalences respect monodromies.
Comment: 41 pages. We have rewritten part of the paper, in particular the introduction in order to make more evident the novelty of our results and added a corollary on the antiequivalence between the category of log $p$-divisible groups and the category of strongly divisible modules of weight less or equal to 1