Bounded Height Conjecture for unlikely intersections in abelian varieties

Let $A$ be an abelian variety of dimension $g$, let $V\subset A$ be an irreducible subvariety of dimension $d$, and let $\Gamma\subset A$ be a subgroup of finite rank. The variety $V$ is weak-transverse if it is not contained in any proper algebraic subgroup of $A$, and transverse if it is not contained in any translate of a proper algebraic subgroup. For a height-induced seminorm, let $\mathcal{O}_\varepsilon=\{\xi\in A:\|\xi\|\leq\varepsilon\}$ and $\Gamma_\varepsilon=\Gamma+\mathcal{O}_\varepsilon$. Define $S_k(V^e,F)=V^e\cap\bigcup_{\operatorname{cod}B\geq k}(B+F)$, with $B$ ranging over abelian subvarieties of codimension at least $k$. Bounded Height Conjecture. There exists $\varepsilon>0$ and a nonempty Zariski-open set $V^0\subset V$ such that:

1. If $V$ is weak-transverse, then $S_{d+1}(V^0,\mathcal{O}_\varepsilon)$ has bounded height.
2. If $V$ is transverse, then $S_{d+1}(V^0,\Gamma_\varepsilon)$ has bounded height.

The conjecture is the height component of the preceding non-density statement. The source notes that the weak-transverse assertion implies the transverse one, whereas the converse does not hold in general.