The period formula for equal parameters in the complex rational delay recurrence

Let $z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma-z_n}$ be the complex rational delay difference equation, where $k$ is the delay and $p$ denotes the period of a periodic solution. Equal-parameter period conjecture. When $\alpha=\beta=\gamma$, the period satisfies

$$p=k+3.$$

The formula is suggested by the tabulated examples, but the supplied text gives no proof and does not specify the admissible initial values or parameter range beyond the equality $\alpha=\beta=\gamma$.