The higher-codimension Bounded Height Conjecture

Let $A$ be a semi-abelian variety, let $V$ be an irreducible algebraic subvariety of dimension $d$ defined over $\overline{\mathbb{Q}}$, and let $\Gamma$ be a finite-rank subgroup of $A(\overline{\mathbb{Q}})$, with $\Gamma_\varepsilon=\Gamma+\mathcal{O}_\varepsilon$. A subvariety is weak-transverse if it is not contained in a proper algebraic subgroup, and transverse if it is not contained in a proper coset. Higher-codimension Bounded Height Conjecture. There exist $\varepsilon>0$ and a non-empty Zariski open subset $V^e$ of $V$ such that:

1. If $V$ is weak-transverse, then $S_{d+1}(V^e,\mathcal{O}_\varepsilon)$ has bounded height.
2. If $V$ is transverse, then $S_{d+1}(V^e,\Gamma_\varepsilon)$ has bounded height.

This extends the bounded-height prediction from codimension $d$ to codimension $d+1$, distinguishing the weak-transverse and transverse cases. The source presents it as a natural question following the optimality result for varieties failing Property $(S)$; its general validity is open.