Zhang's finiteness conjecture for small specializations of sections

Let $A\to T^0$ be a non-isotrivial family of abelian varieties over a number field $K$, with generic fiber $A_{\eta}$ simple of dimension at least two. A section $P:T^0\to A$ is non-torsion if it is not a torsion section, and $h_{A_t}(P_t)$ denotes the canonical height of its specialization at $t$. Zhang's conjecture. For each non-torsion section $P:T^0\to A$ defined over $\overline{\mathbb{Q}}$, there exists an $\epsilon>0$ such that

$$\{t\in T^0(\overline{\mathbb{Q}}): h_{A_t}(P_t)\le\epsilon\}$$

is finite. The conjecture predicts a strong finiteness property for points where a non-torsion section specializes to a point of uniformly small canonical height; it would make simultaneous sequences of small specializations highly constrained. Its status is not resolved in the supplied source context.