Geometric divisibility sequence constancy conjecture for higher-dimensional semiabelian groups

Let $\mathcal{G}/\mathbb{Z}$ be a group scheme, let $\mathcal{P}\in\mathcal{G}(\mathbb{Z})$ be a $\mathbb{Z}$-valued point, and assume that

1. the generic fiber $G=\mathcal{G}\times_{\mathbb{Z}}\mathbb{Q}$ is an irreducible commutative algebraic group of dimension at least $2$ with no unipotent part;
2. if $P\in G(\mathbb{Q})$ is the restriction of $\mathcal{P}$ to the generic fiber, then the subgroup $\mathbb{Z}P$ generated by $P$ is Zariski dense in $G$.

Geometric divisibility sequence conjecture. The geometric divisibility sequence $(D_{n\mathcal{P}})_{n\geq 1}$ corresponding to $\mathcal{P}$ satisfies

$$D_{n\mathcal{P}}=D_{\mathcal{P}}$$

for infinitely many integers $n\geq 1$. This extends the type of recurrence predicted by the Ailon–Rudnick conjecture to higher-dimensional commutative algebraic groups without unipotent parts; the assertion is presented as a conjecture and no resolution is given in the source.