Finite termination of splitting processes for accessible quasi-transitive graphs

Let $G$ be an accessible connected quasi-transitive locally finite graph. A splitting process is any process of repeatedly splitting $G$ as described in the paper.

Finite termination conjecture. Every process of splittings must end after finitely many steps.

The conjecture asks whether the terminal factorisation is independent of the particular choices made during splitting. The paper does not establish this for arbitrary splittings; its preceding construction uses a specific invariant tree-decomposition.