Geometric convergence conjecture for cascade series

Let a non-null probabilised context tree be given, and consider its cascade series. A subtree is shift-stable when it is preserved by the relevant shift operation.

Geometric convergence conjecture. If the tree has no infinite shift-stable subtree, then all its cascade series converge with geometric rates.

The conjecture is motivated by the contrast between the left comb, whose cascade series may converge or diverge depending on the distributions, and the bamboo blossom, whose cascade series converge geometrically for every non-null choice of distributions. The claim remains an open proposed generalization of these examples.