The zero-entry conjecture for satisfying Loosened Collatz tuples

Let a tuple $(y_1,\dots,y_n)$ be satisfying when it satisfies the Loosened Collatz Function, and let $(2,2,\ldots)$ denote the constant tuple with every entry equal to $2$. Zero-entry conjecture. Excluding the tuple $(2,2,\ldots)$, a tuple satisfying the Loosened Collatz Function must include $0$ in its entry. The claim predicts that every nonconstant satisfying tuple contains a zero entry; the source gives no resolution, so it remains open.