Bogomolov's conjecture for non-isotrivial curves over function fields

Let $K$ be a function field or a number field, let $C$ be a smooth projective curve of genus $g\geq 2$ over $\overline{K}$, let $J_C$ be its Jacobian, and fix a degree-one divisor $D$ on $C$. Define

$$B_C(P,\epsilon):=\left\{x\in C(\overline{K})\mid \|\jmath_D(x)-P\|_{NT}\leq \epsilon\right\},$$

where $\jmath_D(x)=x-D$ and $\|\cdot\|_{NT}$ is the seminorm from a Néron–Tate height associated to a symmetric theta divisor. When $K$ is a function field, assume that $C$ is non-isotrivial. Bogomolov's conjecture. For every $P\in J_C(\overline{K})$, there exists $r>0$ such that $B_C(P,r)$ is finite. The conjecture is known for curves over number fields and, as established in this paper, over arbitrary function fields; thus this candidate is solved.