The one-branch conjecture for real Collatz graphs

For each $x\in(0,\infty)$, let $G_x$ be the directed graph constructed from the balanced ternary data associated with $x$. A branch is a directed path in $G_x$ of the type used in the paper’s graph construction. One-branch conjecture. $G_x$ has exactly one branch for a Lebesgue-co-null set of $x\in(0,\infty)$.

The paper proves the analogous assertion for a comeager set of real numbers. The conjecture asks for the corresponding full-measure statement.