Density of orbits of endomorphisms of commutative linear algebraic groups

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon G\longrightarrow G$ is a dominant endomorphism, we prove that one of the following holds: either there exists a non-constant rational function $f\in $\mathbb{k}$(G)$preservedby$\Phi$(i\mathrm{.}e\mathrm{.},$f\circ \Phi = f$), or there exists a point $x\in G(\mathbb{k})$ whose $\mathrm{\Phi }$-orbit is Zariski dense in $G$.