Uniform finiteness conjecture for isogeny classes in one-parameter families

Let $K$ be a number field, $S$ a set of finite places of $K$, and $g\geq 2$ an integer. Let $\mathcal{F}$ be a one-parameter family of complex abelian varieties $A$ of dimension $g$, parametrized by an algebraic curve $\mathcal{C}$ defined over $K$. Let $\mathcal{U}(K,S,\mathcal{F})$ be the set of ($K$-isogeny classes of) abelian varieties $[A]_K$, defined over $K$ and having good reduction outside $S$, such that some $A'\in[A]_K$ satisfies $A'(\mathbb{C})\in\mathcal{F}$. Uniform finiteness conjecture. The quantity $|\mathcal{U}(K,S,\mathcal{F})|$ depends at most on $K$ and $g$. This conjecture seeks to make the Lawrence–Venkatesh approach to uniform boundedness effective across one-parameter families by reducing the problem to uniform finiteness of suitable families of Galois representations; its status is not specified in the source.