On Endomorphisms of the Cuntz Algebra which Preserve the Canonical UHF-Subalgebra, II

It was shown recently by Conti, R{\o}rdam and Szyma\'{n}ski that there exist endomorphisms $\lambda_u$ of the Cuntz algebra $\mathcal{O}_n$ such that $\lambda_u (\mathcal{F}_n)\subseteq\mathcal{F}_n$ but $u\not\in\mathcal{F}_n$, and a question was raised if for such a $u$ there must always exist a unitary $v\in\mathcal{F}_n$ with $\lambda_u|_{\mathcal{F}_n} = \lambda_v|_{\mathcal{F}_n}$. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms $\lambda_u$ for which the relative commutant $\lambda_u(\mathcal{F}_n)'\cap\mathcal{F}_n$ is finite dimensional.