Discriminant-negligibility of heights of special points on Shimura varieties

Let $(\mathbf{G},\mathbf{X})$ be a Shimura datum with $\mathbf{G}$ semisimple of adjoint type, let $K$ be a compact open subgroup of $\mathbf{G}(\mathbb A_f)$, let $X$ be a connected component of $\mathbf{X}$, and set $S=\Gamma\backslash X$, where $\Gamma=K\cap\mathbf{G}(\mathbb Q)^+$. Let $p\in S$ be a special point, write $p=\pi(x)$ with $x$ in a fixed semialgebraic fundamental set, and let $\mathbf{T}$ be the Mumford–Tate torus of $x$. If $K^m_{\mathbf{T}}$ is the maximal compact open subgroup of $\mathbf{T}(\mathbb A_f)$, $K_{\mathbf{T}}=K\cap\mathbf{T}(\mathbb A_f)$, $L$ is the splitting field of $\mathbf{T}$, and $d_L$ is the absolute value of its discriminant, define

$$\operatorname{disc}(p)=[K^m_{\mathbf{T}}:K_{\mathbf{T}}]d_L.$$

Discriminant-negligibility conjecture. There exists a Weil height function $h\colon S(\overline{\mathbb Q})\to\mathbb R$ such that, for every special point $p\in S$ and every $\varepsilon>0$,

$$h(p)=O_{S,\varepsilon}(\operatorname{disc}(p)^\varepsilon).$$

Thus the heights of special points grow sub-polynomially in their discriminants. Assuming this conjecture, the paper derives lower bounds for Galois orbits of special points and a new point-counting proof of the André–Oort conjecture for Shimura varieties of abelian type; the conjecture itself remains open in the supplied text.