Strongest Lang–Vojta exceptional-locus conjecture

Let $k$ be an algebraically closed field of characteristic zero and let $X$ be a projective variety over $k$. Let $\Delta_X^{\mathrm{gr}}$, $\Delta_X^{\mathrm{Mor}}$, $\Delta_X^{\mathrm{geom-hyp}}$, $\Delta_X^{\mathrm{1-bounded}}$, $\Delta_X^{\mathrm{bounded}}$, and $\Delta_X^{\mathrm{alg-hyp}}$ denote the corresponding exceptional loci, and let $\Delta_X^{\mathrm{Br}}$ and $\Delta_X^{\mathrm{Kob}}$ denote the Brody and Kobayashi exceptional loci. Strongest Lang–Vojta conjecture. These loci satisfy

$$\Delta_X^{\mathrm{gr}}=\Delta_X^{\mathrm{Mor}}=\Delta_X^{\mathrm{geom-hyp}}=\Delta_X^{\mathrm{1-bounded}}=\Delta_X^{\mathrm{bounded}}=\Delta_X^{\mathrm{alg-hyp}}.$$

Moreover, $X$ is of general type if and only if $\Delta_X^{\mathrm{gr}}\neq X$, and if $k=\mathbb{C}$ then

$$\Delta_X^{\mathrm{gr}}=\Delta_X^{\mathrm{Br}}=\Delta_X^{\mathrm{Kob}}.$$

The conjecture is the strongest exceptional-locus formulation of Lang–Vojta; the source provides no resolution status.