Intersections of orbits of self-maps with subgroups in semiabelian varieties

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$, endowed with a rational self-map $\mathrm{\Phi }$. Let $\alpha \in G(K)$ and let $\mathrm{\Gamma }\subseteq G(K)$ be a finitely generated subgroup. We show that the set $\{n\in \mathbb{N}\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}^n(\alpha )\in \mathrm{\Gamma }\}$ is a union of finitely many arithmetic progressions along with a set of Banach density equal to $0$. In addition, assuming $\mathrm{\Phi }$ is regular, we prove that the set $S$ must be finite.