Rational self-maps with a regular iterate on a semiabelian variety

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$. Let $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:G⇢G$ be a dominant rational self-map. Assume that an iterate ${\mathrm{\Phi }}^m\text{\hspace{0.17em}⁣}:G\to G$ is regular for some $m⩾1$ and that there exists no non-constant homomorphism $\tau :G\to G_0$ of semiabelian varieties such that $\tau \circ {\mathrm{\Phi }}^{mk}=\tau$ for some $k⩾1$. We show that under these assumptions $\mathrm{\Phi }$ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp.