Period-two conjecture for the complex rational difference equation

Let $(z_n)$ be a solution of the difference equation

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

with arbitrary initial values and parameters $\alpha$ and $\beta$ satisfying $\beta=\alpha+1$. Period-two conjecture. All solutions are periodic of period $2$.

The paper notes that this observation is supported by numerical experiments and is a result when the parameters and initial values are real; the claim for arbitrary complex parameters and initial values is presented as a conjecture.