Conjecture on the parameter condition for chaotic solutions

Consider the difference equation

$$z_{n+1}=\frac{\alpha+\alpha z_n+\beta z_{n-1}}{1+z_n}.$$

Here $\alpha$ and $\beta$ are complex parameters, and a solution is called chaotic in the sense used by the paper's numerical dynamical analysis. Chaotic-solution conjecture. Chaotic solutions exist for the equation if and only if

$$\left\lvert\alpha+1\right\rvert>\left\lvert\beta\right\rvert.$$

The conjecture is based on the observation that all seven numerical examples in the paper's table satisfy this inequality. No proof or resolution is supplied, and the meaning of chaos is tied to the paper's example-based analysis.