Is there someone who can show me how I can prove this conjecture? Or at least show me how to do the first implication ?

Conjecture: Assume $\alpha,\beta, \lambda \in [0,\infty)$. Then every positive solution of the difference equation : $$z_{n+1}=\frac{\alpha+z_{n}\beta +z_{n-1}\lambda}{z_{n-2}},\quad n=0,1,\ldots$$ is bounded if and only if $\beta=\lambda$

Any help is very welcome. Thank you for any comments or any replies.

Edit: as mentioned in the comments, this is conjecture 8 in this paper by Ladas, Lugo and Palladino