The depthwise discrepancy recurrence conjecture for Collatz orbits

Let $\delta_K(n_0)$ denote the total-variation discrepancy at modular depth $K$ for the orbit starting at odd $n_0$. Depthwise discrepancy 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 $K\ge3$,

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

Such a recurrence would give a dynamical mechanism for decay of discrepancies and would imply the weak-mixing bound discussed in the paper. It remains conjectural.