Boundary conjecture for the hyperbolic filtration

Let $\mathfrak H=\{\mathfrak H_\alpha\}_{\alpha\in(0,2/3]}$ be the hyperbolic filtration, with boundary $\partial\mathfrak H_\alpha$ in the relevant function class. For $\alpha\in(0,2/3]$, define

$$f(z)=z\left(\frac{1+z}{1-z}\right)^{\frac{\alpha}{2(1-\alpha)}}.$$

Boundary conjecture. The function $f$ belongs to $\partial\mathfrak H_\alpha$ whenever $\alpha\in(0,2/3]$. The source notes that several other boundary functions are known, while the full boundary of the hyperbolic filtration is more complicated and this assertion remains unproved in the supplied text.