Pseudohyperbolic disk conjecture for unicritical Blaschke products

Let $t\in[0,1)$, and let $\Theta_t$ be of the unicritical form referred to in the source. Let $W(\Theta_t)$ denote the associated numerical range. Pseudohyperbolic disk conjecture. If $n\geq3$, then $W(\Theta_t)$ contains a pseudohyperbolic disk of radius

$$\cos\left(\frac{\pi}{n+1}\right).$$

The conjecture is motivated by evidence for $n=6,7,8,11$ and by a preceding theorem, but the supplied text does not establish it in general. The notation for the unicritical form and the precise meaning of $W(\Theta_t)$ are defined elsewhere in the paper.