Stabilization conjecture for totally non-stable context trees

Let $({\mathscr T},q)$ be a non-null probabilised context tree. A context tree is totally non-stable when its set of infinite branches ${\mathscr C}^i$ has no shift-invariant subset. A context tree is stabilizable when the stabilization construction referred to in the source is defined. Stabilization conjecture. If ${\mathscr T}$ is totally non-stable and stabilizable, then its stabilized tree satisfies the conditions of the cited theorem on the stable case. This is presented as a weaker first step toward the conjecture on uniqueness of the stationary measure, and the supplied text gives no resolution.