Uniform Mordell–Lang conjecture for curves

Let $g\ge 2$ and $r\ge 0$. For a curve $C$ over $\mathbb{C}$ of genus $g$, let $J$ be its Jacobian and let $\iota\colon C\to J$ be an embedding. Let $\Gamma\subset J(\mathbb{C})$ be a subgroup of rank $r$. Uniform Mordell–Lang conjecture for curves. There is a constant $N(g,r)$ such that

$$\#\iota^{-1}(\Gamma)\le N(g,r).$$

For each individual curve and subgroup, finiteness follows from work of Faltings, while heuristic arguments and results over function fields support the existence of a uniform bound; the complex case remains open and is related in the source to a special case of the Zilber–Pink conjecture.