The Orbit Equidistribution Conjecture for Collatz orbits

Let $T$ denote the odd-to-odd Syracuse map, let $n_0$ be odd, and define $a_i=T^i(n_0)\bmod 2^M$. Let $\mu_{\mathrm{orb},N}=\frac{1}{N}\sum_{i=1}^N\delta_{a_i}$ and let $\mu_U$ be the uniform distribution on the admissible residue classes modulo $2^{M(N)}$. The Orbit Equidistribution Conjecture. There exists a function $M(N)\to\infty$ such that

$$\|\mu_{\mathrm{orb},N}-\mu_U\|_{\mathrm{TV}}\to0$$

as $N\to\infty$ modulo $2^{M(N)}$, uniformly in the depth. In particular, fixed-modulus equidistribution follows, and the conjecture would provide the tail control needed to pass from truncated to full orbitwise means. The paper presents this as an unproved orbitwise hypothesis supporting its conditional reductions.