Vanishing-of-cascades conjecture for sigma-finite invariant measures

Let $U$ be a variable-length memory chain defined by a probabilised stable context tree, and suppose that $U$ admits an invariant $\sigma$-finite measure. For every $\alpha s\in{\mathscr S}$, let $\left(\kappa_{\alpha s}(n)\right)_n$ be the associated sequence, where ${\mathscr S}$ is the set of alpha-LIS objects.

Vanishing-of-cascades conjecture. For every $\alpha s\in{\mathscr S}$, the sequence $\left(\kappa_{\alpha s}(n)\right)_n$ tends to $0$ as $n$ tends to infinity. Consequently, $Q$ is stochastic.

For stable context trees, vanishing of cascades already implies that $Q$ is stochastic, while convergence of cascade series is known to be necessary for an invariant probability measure. The conjecture proposes the analogous necessity of vanishing cascades for the existence of an invariant $\sigma$-finite measure.