Non-density Conjecture for bounded-height subsets of 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 $\mathcal{O}_\varepsilon=\{\xi\in A:\|\xi\|\leq\varepsilon\}$, $\Gamma_\varepsilon=\Gamma+\mathcal{O}_\varepsilon$, and $V_{K_0}=V\cap\mathcal{O}_{K_0}$. Define $S_k(V^e,F)=V^e\cap\bigcup_{\operatorname{cod}B\geq k}(B+F)$. For all real $K_0$, there exists an effective $\varepsilon>0$ such that:

1. If $V$ is weak-transverse, then $S_{d+1}(V_{K_0},\mathcal{O}_\varepsilon)$ is not Zariski-dense in $V$.
2. If $V$ is transverse, then $S_{d+1}(V_{K_0},\Gamma_\varepsilon)$ is not Zariski-dense in $V$.

The source proves that the two parts are equivalent. This conjecture would imply effective Bogomolov-type statements, and its transverse part also implies a Mordell–Lang plus Bogomolov theorem.