Uniform boundedness conjecture for rational points on curves over complex function fields

Let $L={\mathbb C}(B)$ be a complex function field and let $g\geq 2$. Let ${\mathcal C}_g(L)$ denote the set of nonsingular projective curves of genus $g$ defined over $L$ as in the source.

Uniform boundedness conjecture. For any complex function field $L$ and any integer $g\geq 2$ there exists a number $B_g(L)$ such that for every $C\in{f{\mathcal C}_g(L)}$ one has

$$|C(L)|\leq B_g(L).$$

This is the function-field analogue of the uniformity questions discussed over number fields. The source gives no resolution status.