The consequence of the extended Lang–Vojta conjecture for pointed maps

Let $U$ be a smooth quasi-projective variety, and let $Z\subsetneq U$ be a proper closed subset such that every non-constant holomorphic map $\mathbb{C}\to U^{\operatorname{an}}$ factors over $Z^{\operatorname{an}}$. For a variety $C$, let $\operatorname{Hom}(C,U)$ denote the moduli space of morphisms from $C$ to $U$, let $\operatorname{Hom}^{nc}(C,U)$ denote its non-constant locus, and similarly for $\Delta\subset U$.

Consequence of the extended Lang–Vojta conjecture. There is a proper closed subset $\Delta\subsetneq U$ such that:

1. $U$ is bounded modulo $\Delta$.
2. Every non-constant holomorphic map $\mathbb{C}\to\operatorname{Hom}(C,U)^{\operatorname{an}}$ factors over $\operatorname{Hom}(C,\Delta)^{\operatorname{an}}$.
3. For every smooth quasi-projective curve $C$,

$$\dim\operatorname{Hom}^{nc}(C,U)\setminus\operatorname{Hom}(C,\Delta)\leq\dim X-1.$$

4. For every variety $Y$, every point $y$ of $Y$, and every $u\in U\setminus\Delta$, the set of morphisms $f:Y\to U$ satisfying $f(y)=u$ is finite.

The source states that this conjecture follows from the extended Lang–Vojta conjecture. It packages boundedness, hyperbolicity inheritance, a dimension bound for non-constant maps, and finiteness of pointed maps, but no resolution is supplied.