Lang-Vojta conjecture over function fields for very general log projective spaces

We consider a bound of degrees of curves by their genera in the log pair $({\mathbb{P}}^n\times C,D)$, where $C$ is a smooth complex projective curve and $D$ is a very general smooth divisor on ${\mathbb{P}}^n\times C$. Our study is motivated by a function field analogue of the Lang-Vojta conjecture.