The gain-observable recurrence conjecture for Collatz orbits

Let $\eta_K(n_0)$ be the phantom gain discrepancy at depth $K$ for the odd-to-odd Syracuse orbit starting at odd $n_0$. Gain-observable recurrence conjecture. There exist constants $\lambda\in(0,1)$, $C>0$, and $\beta>1$ such that, for every odd starting value $n_0$ and every $K\ge3$,

$$\eta_{K+3}(n_0)\le\lambda\,\eta_K(n_0)+C K^{-\beta}.$$

Controlling the gain observable is a sharper target than controlling the full total-variation discrepancy and is proposed as a potentially more tractable route to the weak-mixing estimate. The conjecture is open.