Ringel's cluster-additive function conjecture for Dynkin diagrams

Let $\Gamma = \mathbb{Z} \Delta$, where $\Delta$ is one of the Dynkin diagrams $\mathbb{A}_n$, $\mathbb{D}_n$, $\mathbb{E}_6$, $\mathbb{E}_7$, and $\mathbb{E}_8$, and let $f$ be cluster-additive on $\Gamma$. A function on $\Gamma$ is cluster-additive if it satisfies the cluster-additivity condition defined in the source. Ringel's conjecture. The function $f$ is a non-negative linear combination of cluster-hammock functions; equivalently, there are a tilting set $\mathcal{T}$ and integers $n_x \in \mathbb{N}_0$ for all $x \in \mathcal{T}$ such that

$$f = \sum_{x \in \mathcal{T}} n_x h_x.$$

This conjecture characterizes cluster-additive functions on repetition quivers of Dynkin type in terms of cluster-hammock functions. The paper presents it as the conjecture whose proof is developed in the surrounding section; the supplied text does not establish its resolution, so its status is recorded as open.