Grove–Ladas non-equilibrium limit conjecture for a max-ratio recurrence

Let $A\in(0,\infty)$, and consider a positive solution $(x_n)$ of the recurrence

$$x_{n+1}=\frac{\max\{A,x_n,x_{n-1}\}}{x_{n-2}}.$$

A solution is non-equilibrium if it is not an equilibrium solution of this recurrence.

Grove–Ladas conjecture. No positive non-equilibrium solution has a limit; the question is also to extend and generalize this assertion.

This conjecture was proposed by Grove and Ladas and is cited here as a related question for the transformed max-type equation. The statement concerns asymptotic behavior rather than periodicity, and its status is not resolved in the supplied text.