Edixhoven's Galois-orbit conjecture for special points

Let $(G,X)$ be a Shimura datum, let $K$ be neat and a product of compact open subgroups $K_p$ of $G(\mathbb Q_p)$, and fix a connected component $S=\Gamma\backslash X^+$, where $\Gamma=G(\mathbb Q)_+\cap K$. For a special point $[h]\in S$, let $M=\operatorname{MT}(h)$ be its Mumford–Tate torus, let $L$ be its splitting field, let $K_M=M(\mathbb A_f)\cap K$, let $K_M^m$ be the maximal compact open subgroup of $M(\mathbb A_f)$, let $i(M)$ be the number of places $p$ with $K_{M,p}\ne K^m_{M,p}$, and let $D_L$ be the absolute value of the discriminant of $L$. Edixhoven's Galois-orbit conjecture. There exist positive constants $c_1$, $B_1$ and $\mu_1$ such that, for every special point $[h]\in S$,

$$|\operatorname{Gal}(\overline{\mathbb Q}/L)\cdot[h]|>c_1B_1^{i(M)}[K^m_M:K_M]D_L^{\mu_1}.$$

Such lower bounds are intended as the arithmetic input for the André–Oort strategy, but the supplied text gives no resolution of this conjecture.