Correlation conjecture for stably integral points on quasi-abelian subvarieties

Let $(A\to B,\Theta)$ be a family of smooth principally polarized quasi-abelian varieties over a number field $K$. Let $H\subset A$ be a family of quasi-abelian subvarieties. Let ${\mathcal P}\subset A(K)$ be the set of stably integral points.

Correlation conjecture. The intersection ${\mathcal P}\cap H$ is correlated with respect to the morphism $H\to B$.

This conjecture is proposed as the reduction needed for extending the argument to higher dimensions and to arbitrary polarizations. The supplied text gives no resolution or further evidence for it.