Right-eigenvalue conjecture for the context-tree matrix

Let a probabilised context tree be given, with associated square matrix $Q$, and let ${\mathscr S}$ denote the set of alpha-LIS objects. For each $\alpha s\in{\mathscr S}$, let $\left(\kappa_{\alpha s}(n)\right)_n$ be the associated sequence.

Right-eigenvalue conjecture. If, for every $\alpha s\in{\mathscr S}$, the sequence $\left(\kappa_{\alpha s}(n)\right)_n$ converges to $0$, then $Q$ admits $1$ as a right eigenvalue.

When the context tree is stable, the matrix $Q$ is stochastic under the same vanishing condition. The conjecture asks whether the right eigenvalue still exists without stability; together with the stated correspondence between right eigenvectors and invariant measures, it would imply existence of an invariant probability measure when the set of alpha-LIS is finite and all cascade series converge.