Strong Lang–Vojta conjecture II

Let $k$ be an algebraically closed field of characteristic zero. For a projective variety $X$ over $k$, consider pseudo-Mordellicity, pseudo-arithmetic hyperbolicity, pseudo-grouplessness, and being of general type. Strong Lang–Vojta conjecture II. These four properties are equivalent. This extends the conjecture from $\overline{\mathbb{Q}}$ to arbitrary algebraically closed fields of characteristic zero; the source provides no resolution status.