Silverman's conjecture on geometric divisibility sequences

Let $\mathcal{G}/\mathbb{Z}$ be a group scheme, let $\sigma_P\in\mathcal{G}(\mathbb{Z})$, and write

$$G=\mathcal{G}\times_{\mathbb{Z}}\mathbb{Q}$$

for its generic fiber. Assume that $G$ is an irreducible commutative algebraic group of dimension at least $2$ with no unipotent part, and that the restriction $P\in G(\mathbb{Q})$ of $\sigma_P$ to the generic fiber generates a Zariski-dense subgroup $\mathbb{Z}\cdot P$ in $G$. Let $(D_{nP})_{n\geq 1}$ be the geometric divisibility sequence corresponding to $\sigma_P$. Silverman's conjecture. One has

$$D_{nP}=D_P$$

for infinitely many $n\geq 1$. This generalizes the Ailon–Rudnick phenomenon from sequences of the form $a^n-1$ to geometric divisibility sequences arising from group schemes. The paper presents this as a conjecture and subsequently studies cases in which it can be proved.