Vojta's height conjecture for families of higher-genus curves

Let ${\mathcal C}\to B$ be a family of higher-genus curves, let $b\in B({\mathbb Q})$, and let ${\mathcal C}_b$ be a smooth fiber. Let $H_B$ be a non-logarithmic height on $B$, and let $H_{\mathcal C}$ be a suitable height function on ${\mathcal C}$. Vojta's height conjecture. There are constants $\gamma$ and $\kappa$ such that

$$H_{\mathcal C}(P)\leq\gamma H_B(b)^\kappa$$

for all $P\in {\mathcal C}_b({\mathbb Q})$. This is presented as a consequence of Vojta's conjecture and gives a uniform polynomial height bound for rational points in smooth fibers; the source does not state a resolution.