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][1]


  [1]: http://anubih.ba/Journals/vol.8,no-2,y12/11Ladas-Lugo-Palladino.pdf