Unique stationary measure 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. Unique stationary-measure conjecture. If ${\mathscr T}$ is totally non-stable, then there exists a unique probability stationary measure for the VLMC associated to $({\mathscr T},q)$. The conjecture concerns existence and uniqueness of stationary laws in the non-stable regime; the supplied text presents it as a proposed direction and gives no resolution.