The peripheral-spectrum conjecture for unital completely positive maps

Let $\Phi:{\mathfrak A}\to{\mathfrak A}$ be a unital completely positive map acting on the unital $C^*$-algebra ${\mathfrak A}$, and let $\sigma(\Phi)$ denote its spectrum. Peripheral-spectrum conjecture. If

$$\sigma(\Phi)\subset{\mathbb T},$$

then $\Phi$ is a $*$-automorphism. This would characterize $*$-automorphisms among unital completely positive maps whose spectrum lies on the unit circle; the supplied text does not indicate whether the assertion is known or remains open.