Poonen's Mordell–Lang–Bogomolov conjecture for semiabelian varieties

Let $A$ be a semiabelian variety over a number field $k$, let $\Gamma$ be a finitely generated subgroup of $A(\overline{k})$, and let

$$\Gamma' = \{x \in A(\overline{k}) \mid nx \in \Gamma \text{ for some } n \geq 1\}$$

be its division group. Let $X$ be a geometrically integral closed subvariety of $A$. A sequence in $A(\overline{k})$ is a sequence of small points if it is small for the dynamical height construction associated with multiplication on $A$.

Poonen's Mordell–Lang–Bogomolov conjecture. For $i \geq 1$, suppose

$$x_i=\gamma_i+z_i \in X(\overline{k}),$$

where $\gamma_i \in \Gamma'$ and $\{z_i\}_{i \geq 1}$ is a sequence of small points in $A(\overline{k})$. If $X$ is not a translate of a sub-semiabelian variety of $A$ by an element of $\Gamma'$, then the points $x_i$ are not Zariski dense in $X_{\overline{k}}$.

The conjecture extends Mordell–Lang and Bogomolov from abelian to semiabelian varieties. The source presents it as unresolved, while noting that many special cases were already known; when $\Gamma=0$, it gives a semiabelian generalization of Bogomolov.