Uniform Mordell–Lang conjecture for rational points on curves

Let $d\ge 1$, $g\ge 2$, and $r\ge 0$. Let $K$ be a number field of degree $d$, and let $C$ be a curve over $K$ of genus $g$ with Jacobian $J$. Uniform Mordell–Lang conjecture for rational points on curves. There is a constant $R(d,g,r)$ such that, whenever $\operatorname{rank}J(K)=r$,

$$\#C(K)\le R(d,g,r).$$

This is presented as a weaker arithmetic variant of the geometric uniform Mordell–Lang statement. The paper proves a uniform bound for hyperelliptic curves in the range of Mordell–Weil rank relevant to its main theorem, but this general arithmetic formulation is not resolved there.