The zero-axis conjecture for dual objects of Loosened Collatz cycles

Let a satisfying cycle determine its dual object in Euclidean space by interpreting its associated tuple and its bitwise rotations as vertices in $\mathbb{R}^n$. A vertex is said to be zero on an axis when at least one of its coordinates is $0$. Zero-axis conjecture. Excluding the point $(2,2,\ldots)$, all dual objects of satisfying cycles require their vertices to be $0$ on at least one axis. The source presents this as equivalent to the preceding tuple conjecture and gives no resolution, so it remains open.