@fedja, the problem is open is not as what you think just a question .

@fedja .This conjecture need many experts in dynamical system and bifurction to proof it. I don't know if i accept this answer or should be waite decision of experts in this field .

@losif Pinelis :thank you for your this good answer.It seems the "if only" second part necessite a huge work .

j.Diaz , thank you for your answer but you take only k as odd integer, what about k is even ?

i think recursive formula

@GHfromMO, i knew this that's published conjecture after positing it , in the fisrt it was my independent work

@GHfromMO you can edit your answer

@JosephO'Rourke, only what i have that this conjecture deal with the following simpler system: $$\begin{cases} x_{n+1}=\dfrac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\dfrac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\dfrac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{n}}{A_{3}+B_{3}x_{n}+c_{3}y_{n}+D_{3}z_{n}}{}\end{cases} \quad n=0,1,\dots,$$ .with non-negative paramaters and non-negative initial conditions and dominators are never zero

no, i'm not intended any specific initial values