Lang's intermediate conjecture for exceptional loci

Let $k$ be an algebraically closed field of characteristic zero, let $X$ be a projective variety over $k$, let $\Delta\subset X$ be a closed subset, and let $p$ be a positive integer. The paper defines Mordellicity, algebraic hyperbolicity, and algebraic boundedness modulo $\Delta$. Lang's intermediate conjecture for exceptional loci. The following statements are equivalent: every subvariety $Y$ of $X$ of dimension at least $p$ with $Y\not\subset\Delta$ is of general type; $X$ is Mordellic modulo $\Delta$ over $k$; $X$ is $p$-Mordellic modulo $\Delta$ over $k$; $X$ is $p$-algebraically hyperbolic modulo $\Delta$ over $k$; and $X$ is $p$-algebraically bounded modulo $\Delta$ over $k$. The source notes that some implications and special cases are known, while the full equivalence is conjectural.