The extended Lang–Vojta conjecture for quasi-projective varieties

Let $U$ be a quasi-projective variety. Write $U^{\operatorname{an}}$ for its associated complex-analytic space. A variety is pseudo-Kobayashi hyperbolic if it has the property specified by that notion in the source, and $U$ is bounded modulo a proper closed subset $\Delta\subset U$ when the boundedness condition in the source holds.

Extended Lang–Vojta conjecture. The following conditions are equivalent:

1. $U$ is pseudo-Kobayashi hyperbolic.
2. There is a proper closed subset $\Delta\subset U$ such that every entire curve $\mathbb{C}\to U^{\operatorname{an}}$ factors over $\Delta^{\operatorname{an}}$.
3. There is a proper closed subset $\Delta\subset U$ such that, for every variety $Y$, every point $y$ of $Y$, and every $x\in U\setminus\Delta$, the set of morphisms $f:Y\to U$ satisfying $f(y)=x$ is finite.
4. $U$ is of log-general type.
5. There is a proper closed subset $\Delta\subset U$ such that $U$ is bounded modulo $\Delta$.

This is proposed as a Lang–Vojta-type conjecture for quasi-projective varieties and is stated to be not new when $U$ is projective; the equivalence of these five conditions remains conjectural in the source.