The summable alignment-renewal conjecture for Collatz shadows

Let $\eta_K^{\mathrm{renew}}(n_0)$ denote the total discrepancy contribution from renewed depth-$K$ alignment events for an orbit of the odd-to-odd Syracuse map starting at odd $n_0$. Summable alignment-renewal conjecture. There exist constants $C'>0$ and $\beta>1$, independent of $n_0$, such that for every odd $n_0$ and every $K\ge3$,

$$\eta_K^{\mathrm{renew}}(n_0)\le C'K^{-\beta}.$$

This conjecture is one of the component estimates proposed to establish summability of the depthwise discrepancies after known-zone memory loss. It is unproved.