Consider the Cuntz algebra $\mathcal{O}_n$ with $n \geq 2$ and let $\text{End}(\mathcal{O}_n)$ be the set of all (unital) $\ast$-endomorphisms of $\mathcal{O}_n$. I was wondering if there exists an element $x \in \mathcal{O}_n$ such that the evaluation map $\text{End}(\mathcal{O}_n) \rightarrow \mathcal{O_n},$ $\phi \mapsto \phi (x)$ is injective.

If no, what is the smallest $k \in \mathbb{N}$ for which $x\in \mathcal{O}_n \otimes \mathbb{C}^k$ exists such that the map $\text{End}(\mathcal{O}_n) \rightarrow \mathcal{O}_n \otimes \mathbb{C}^k$ given by $\phi \mapsto (\phi \otimes \mathrm{id}) (x)$ is injective? Is it $k=n-1$?