A general endomorphism of $\mathbb{P}^k$ has trivial iterated centralizer

Fix integers $k,d\geq2$. Let $\operatorname{End}_d^k$ denote the parameter space of holomorphic endomorphisms of $\mathbb{P}^k$ of algebraic degree $d$. We prove that there exists a dense Zariski open subset $U_{d,k}\subset\operatorname{End}_d^k$, defined over $\mathbb{Q}$, such that, for every $f\i