Breuil's classification conjecture for $p$-divisible groups and finite flat group schemes

Let $k$ be a perfect field of characteristic $p$, let $W(k)$ be its ring of Witt vectors, let $K_0=W(k)[1/p]$, and let $K$ be a finite totally ramified extension of $K_0$ with ring of integers $\mathscr{O}_K$. Set $\mathfrak{S}=W(k)[[u]]$, equipped with Frobenius $\varphi(u)=u^p$. Write $\underline{\operatorname{Mod}}_{\mathfrak{S}}(\varphi)^{\leqslant 1}$ for the category of Kisin modules of height at most $1$, and $(\operatorname{Mod}/\mathfrak{S})^{\leqslant1}$ for the corresponding torsion category. Let $\boldsymbol{\mathcal{G}}_{K_\infty}=\operatorname{Gal}(\overline K/K_\infty)$, and let $T^*_{\mathfrak{S}}$ denote the associated contravariant functors. Breuil's conjecture. There exists an exact anti-equivalence of categories

$$G\leftrightsquigarrow\mathfrak{M}_G$$

between $p$-divisible groups over $\mathscr{O}_K$ and $\underline{\operatorname{Mod}}_{\mathfrak{S}}(\varphi)^{\leqslant1}$, compatible with contravariant Dieudonné crystals and inducing a natural $\boldsymbol{\mathcal{G}}_{K_\infty}$-equivariant isomorphism $T_p(G)\cong T^*_{\mathfrak{S}}(\mathfrak{M}_G)$. Likewise, there exists an exact anti-equivalence

$$H\leftrightsquigarrow\mathfrak{M}_H$$

between $p$-power order finite flat group schemes over $\mathscr{O}_K$ and $(\operatorname{Mod}/\mathfrak{S})^{\leqslant1}$, compatible with contravariant Dieudonné crystals and inducing a natural $\boldsymbol{\mathcal{G}}_{K_\infty}$-equivariant isomorphism $H(\overline K)\cong T^*_{\mathfrak{S}}(\mathfrak{M}_H)$. The paper proves this conjecture, including the compatibility with crystalline Dieudonné theory and the associated Galois representations.