Uniform p-adic proximity conjecture for canonical-lift torsion points

Let ${\mathbb C}_p$ be the completion of the algebraic closure of ${\mathbb Q}_p$, with $v$ the extension of the $p$-adic valuation, and let

$${\mathcal O}={\mathcal O}_{{\mathbb C}_p}:=\{x\in{\mathbb C}_p\mid v(x)\geq 0\}$$

be its ring of integers. Let $Z\subseteq{\mathcal X}_{{\mathbb C}_p}$ be a closed subvariety, and let $\lambda(Z,\cdot)$ be a $p$-adic proximity function for $Z$. Uniform p-adic proximity conjecture. There is a rational number $q\in{\mathbb Q}$ such that, for every point $\xi\in{\mathcal X}({\mathbb C}_p)$ that is a torsion point of a canonical-lift fibre, either $\xi\in Z({\mathbb C}_p)$ or

$$\lambda(Z,\xi)\leq q.$$

The preceding argument establishes such a uniform bound when $Z$ is special; the conjecture asserts that this restriction on $Z$ is unnecessary.