Is there someone show me how I can proof this conjecture at least show me how i can doing

the first implication ?

conjecture :Assume $\alpha,\beta , \lambda \in [0,\infty) $ Then every positive solution of

the difference equation :

$z_{n+1}=\frac{\alpha+z_{n}\beta +z_{n-1}\lambda}{z_{n-2}}$,$n=0,1,....$

is bounded if and only if : $\beta=\lambda$

any help is very welcom , thank you for any comments or any replies .