Chai–Oort Lie algebra characterization of strongly Tate-linear deformation spaces

Let $Y$ be the object used to define the formally smooth deformation space $\mathbf{Def}_{\mathrm{sus}}(Y)$, and let $\mathfrak{a}_Y$ be its ambient Lie algebra. Let $\mathcal{M}_Z$ denote the object attached to a closed subspace $Z\xhookrightarrow{}\mathbf{Def}_{\mathrm{sus}}(Y)$, and let $G(\mathcal{M}_Z)$ be its associated group. For an $F$-stable Lie subalgebra $\mathfrak{b}\subset\mathfrak{a}_Y$, let $Z(\mathfrak{b}^+)\xhookrightarrow{}\mathbf{Def}_{\mathrm{sus}}(Y)$ be the corresponding strongly Tate-linear subspace. Chai–Oort Lie algebra conjecture. For every closed immersion $Z\xhookrightarrow{}\mathbf{Def}_{\mathrm{sus}}(Y)$ and every $F$-stable Lie subalgebra $\mathfrak{b}\subset\mathfrak{a}_Y$,

$$Z\subseteq Z(\mathfrak{b}^+)\quad\Longleftrightarrow\quad \operatorname{Lie}G(\mathcal{M}_Z)\subseteq\mathfrak{b}.$$

In particular,

$$\operatorname{Lie}G(\mathcal{M}_{Z(\mathfrak{b}^+)})=\mathfrak{b}.$$

The preceding discussion identifies $Z(\mathfrak{b}^+)$ with the deformation space of a trivial torsor and calls it strongly Tate-linear in the sense of Chai--Oort; the supplied excerpt does not establish whether this characterization is proved.