Conjecture on fractal-like and fractal-unlike chaotic trajectories

Let

$$f_{\alpha,\beta,\gamma,\delta}(z)=\frac{\alpha z+\beta}{\gamma z^2+\delta z}$$

be the rational map associated with parameters $\alpha$, $\beta$, $\gamma$, and $\delta$. A chaotic trajectory is called fractal-like or fractal-unlike according to the observed geometric behavior. Fractal-trajectory conjecture. The chaotic trajectories of $f_{\alpha,\beta,\gamma,\delta}$ are fractal-like only if

$$|\alpha|<|\gamma|,\qquad |\beta|<|\delta|,\qquad |\alpha+\beta|<|\gamma+\delta|,$$

and fractal-unlike only if

$$|\alpha|>|\gamma|,\qquad |\beta|>|\delta|,\qquad |\alpha+\beta|>|\gamma+\delta|.$$

The claim is based on computational observations of chaotic trajectories and a further sample of 500 trajectories; it is not accompanied by a proof in the supplied text.