The bounded height conjecture for the open anomalous locus

Let $G$ be a group variety, and let $V\subset G$ be irreducible. Let $V^{oa}$ be the set of points of $V$ outside the $V$-anomalous varieties, and define

$$\mathcal{B}_{\dim V}=\bigcup_{\operatorname{codim} B\geq \dim V}B,$$

where the union is over all algebraic subgroups $B$ of $G$ of codimension at least $\dim V$. Bounded height conjecture. The set $V^{oa}\cap\mathcal{B}_{\dim V}$ has bounded height. This generalizes the bounded height conjecture of Bombieri, Masser and Zannier and is intended to address the height obstruction in effective proofs of the torsion anomalous and intersection conjectures.