Green–Griffiths–Lang conjecture for compactified ball quotients

Let \overline{X} = \overline{{\left.\raisebox{-.2em}{\Gamma}\middle\backslash\raisebox{.2em}{\mathbb B^n}\right.}} be a compactification as in Theorem, with $n \geq 6$. Here $D$ denotes the boundary of $\overline{X}$, and an entire curve means a holomorphic map from $\mathbb C$ to $\overline{X}$. Green–Griffiths–Lang conjecture for compactified ball quotients. There exists an algebraic subset $\Sigma \subset \overline{X}$ with

$$\dim \Sigma \leq 5$$

such that every non-constant entire curve, or every subvariety that is not of general type, is contained in $D \cup \Sigma$. This is presented as a consequence of the Green–Griffiths–Lang conjecture in the setting of compactified ball quotients; the stated result uses the preceding theorem that subvarieties of dimension at least $6$ outside the boundary are of general type. The general conjectural statement remains open.