Toeplitz Kernel Question

Question

$$\newcommand{\kern}{\operatorname{Ker}}$$When looking at a Toeplitz Operator (on $\mathbb{H}^2(\mathbb{C^+}))$ with unimodular symbol, $\phi=e^{i \psi}$, what can we say about $\kern T_{\phi}$ when $$\limsup_{x \rightarrow -\infty} \psi (x) - \liminf_{x \rightarrow \infty} \psi (x) < \pi$$ and $$\limsup_{x \rightarrow -\infty} -\psi (x) - \liminf_{x \rightarrow \infty} -\psi (x) < \pi?$$ Page 16 of this paper says this is a basic property that implies that $\kern T_{\phi}$ and $\kern T_{\bar \phi}$ are trivial; however, I cannot find a simple explanation for this. What I feel is a possible explanation is Theorem 5 (part 4) of this work of Hruschev, Nikolski and Pavlov but this hardly seems "basic".