The ergodicity conjecture for the 3x+1 conjugacy map

Let $Q$ be the 3x+1 conjugacy map on $\mathbb Z_2$, and let the ergodic domain of $Q$ be the union of the invariant sets on which $Q$ is ergodic. Ergodicity Conjecture. The ergodic domain of $Q$ has full 2-adic measure. The paper reports that the currently identified ergodic sets have total measure greater than $0.96$, and says that the conjecture is related to the poorly understood distribution of periodic orbits.