Existence of arbitrarily high-order periods for 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, with parameters $\alpha$, $\beta$, $\gamma$ and delay $k$. A period $p$ is the length of a periodic solution of this recurrence. Higher-order-period conjecture. There exist as many higher-order periods of the difference equation as are demanded. The claim is motivated by numerical examples exhibiting periods associated with the delay term; the supplied text does not state a proof or a precise range of parameters.