The function-field analogue of Vojta's stronger conjecture

Let $F$ be an algebraically closed field of characteristic zero, let $C$ be a smooth projective curve over $F$, let $\mathcal{X}$ be a smooth projective variety over $F$, and let $\pi:\mathcal{X}\to C$ be a surjective morphism with connected fibers. Let $D$ be a normal crossing divisor on $\mathcal{X}$, and let $S$ be a finite set of closed points of $C$. For an irreducible curve $Y$ on $\mathcal{X}$ mapping onto $C$, let $\nu_Y:\widetilde{Y}\to Y$ be its normalization, and use the source's definitions of $N_S^{(1)}(D,Y)$, $d_C(Y)$, and $h_{\mathcal{M}}(Y)$.

Function-field Vojta conjecture. Let $\mathcal{A}$ be an ample line bundle on $\mathcal{X}$ and let $r$ be a positive integer. For every $\epsilon>0$, there is a proper Zariski-closed subset $Z$ of $\mathcal{X}$ such that, for all $c\in\mathbb{R}$,

$$N_S^{(1)}(D,Y)+d_C(Y)\geq h_{\mathcal{K}(D)}(Y)-\epsilon h_{\mathcal{A}}(Y)-c$$

holds for all irreducible curves $Y$ on $\mathcal{X}$ not contained in $Z\cup D$ with $\deg((\pi|_Y)\circ\nu_Y)\leq r$, except for a bounded family of curves on $\mathcal{X}$.

This is stated as a function-field analogue of Vojta's stronger conjecture. Its status is not specified in the source.