Furstenberg-type periodicity conjecture for even parameters

Let $E$ denote the set of even integers, let $k\in E$, and let $n$ be an integer. The periodicity conjecture. The $n$-orbit is periodic. The preceding analysis shows divergence for odd $k$ except at specified fixed points, while the behavior for even $k$ is suggested by the examples; establishing periodicity for every integer $n$ remains open.