The spectral diffusion conjecture for Collatz orbit measures

Let $\mu_K$ be the orbit measure on $\mathbb{Z}/2^K\mathbb{Z}$, let $\operatorname{hw}(\xi)$ denote Hamming weight, and define the spectral content at weight $w$ by

$$\mathcal{S}_w(K)=\frac{1}{\binom Kw}\sum_{\operatorname{hw}(\xi)=w}|\widehat{\mu}_K(\xi)|^2.$$

Spectral diffusion conjecture. For every convergent Collatz orbit with odd-step density $\rho<1/\log_2 3$, there is $C(\rho)>0$ such that, for every $w\ge1$,

$$\mathcal{S}_w(K)\le C(\rho)2^{-\alpha_w K},$$

where $\alpha_w>0$ depends only on $w$. The conjecture formalizes decay of Walsh spectral content with modular depth; the supplied text gives supporting numerical observations but no proof.