Mazur's uniform Mordell–Lang conjecture for curves

Let $F$ be a field of characteristic $0$, and let $C$ be a smooth curve over $F$, meaning a geometrically irreducible, smooth, projective curve. Write $\operatorname{Jac}(C)$ for its Jacobian. For a point $P_0\in C(F)$, view $C-P_0$ as a curve in $\operatorname{Jac}(C)$ via the Abel–Jacobi map based at $P_0$.

Mazur's uniform Mordell–Lang conjecture. Let $g\ge 2$ be an integer. Then there exists a constant $c(g)\ge 1$ such that, for every smooth curve $C$ of genus $g$ defined over $F$, every $P_0\in C(F)$, and every subgroup $\Gamma\subseteq\operatorname{Jac}(C)(F)$ of finite rank $\operatorname{rk}(\Gamma)$,

$$\#\bigl((C(F)-P_0)\cap\Gamma\bigr)\le c(g)^{1+\operatorname{rk}(\Gamma)}.$$

This is a uniform form of the Mordell–Lang conjecture for curves, asserting a bound depending only on the genus and the rank of the subgroup. The paper identifies it as a question posed by Mazur and explains that the relative Bogomolov conjecture implies it, but the supplied text does not establish the conjecture independently or give a resolution status.