Caporaso–Harris–Mazur correlation conjecture for varieties of general type

Let $f:X \longrightarrow B$ be a proper morphism of integral varieties whose general fiber is an integral variety of general type. For a sufficiently large integer $n$, consider the $n$-fold fiber product $X^n_B$. Correlation Conjecture. Then $X^n_B$ admits a dominant rational map $h$ to a variety $W$ of general type such that the restriction of $h$ to a general fiber of $f^n$ is generically finite.

The paper proves this statement when the general fiber is an integral surface of general type; the conjecture is the broader form posed by Caporaso, Harris, and Mazur and is connected to geometric and number-theoretic consequences of Lang-type conjectures.