Lang–Vojta's pseudo-algebraic-hyperbolicity conjecture

Let $k$ be an algebraically closed field of characteristic zero. A projective variety $X$ over $k$ is pseudo-algebraically hyperbolic over $k$ if there is a proper closed subset $\Delta\subsetneq X$ such that, for every ample line bundle $L$ on $X$, there is a real number $\alpha(X,\Delta,L)$ depending only on $X$, $\Delta$, and $L$ such that, for every smooth projective connected curve $C$ over $k$ and every morphism $f:C\to X$ with $f(C)\not\subset\Delta$,

$$\deg_C f^*L\leq\alpha(X,\Delta,L)\cdot\operatorname{genus}(C).$$

A projective variety $X$ over $k$ is of general type if, for every irreducible component $Y$ of $X$ and every resolution of singularities $\widetilde{Y}\to Y$, the canonical bundle $\omega_{\widetilde{Y}}$ is big.

Lang–Vojta's pseudo-algebraic-hyperbolicity conjecture. A projective integral variety $X$ over $k$ is of general type if and only if $X$ is pseudo-algebraically hyperbolic over $k$.

The conjecture strengthens pseudo-$1$-boundedness by requiring the degree bound to grow linearly with the genus of the source curve. The source describes this as conjecturally equivalent to the preceding boundedness condition and gives no resolution status.