Zhang's small-height conjecture for simple families of abelian varieties

Let $B$ be a quasiprojective smooth algebraic curve defined over $\overline{\mathbb{Q}}$. Suppose $A \to B$ is a non-isotrivial family of abelian varieties with fiber dimension $>1$, defined over $\overline{\mathbb{Q}}$ with a simple generic fiber. Let $\mathcal{L}$ be a line bundle on $A$ which restricts to an ample and symmetric line bundle on each fiber $A_t$, and let $\hat{h}_t$ be the induced Néron-Tate canonical height on $A_t$, for each $t\in B(\overline{\mathbb{Q}})$. For each non-torsion section $P:B\to A$ defined over $\overline{\mathbb{Q}}$, there is a constant $c=c(\mathcal{L},P)>0$ such that

$$\{t\in B(\overline{\mathbb{Q}}):\hat{h}_t(P_t)<c\}$$

is finite.

Zhang's conjecture. The asserted finiteness holds for every such family, line bundle, and non-torsion section.

The conjecture is a height-theoretic analogue of the Bogomolov phenomenon for non-isotrivial families of abelian varieties. The stated conclusion is known to fail when the abelian variety is not simple, and in particular for families of elliptic curves, so this formulation is refuted.