Component-cluster cardinality and rigidity conjecture for basic algebras

Let $\Lambda$ be a basic algebra with $n$ vertices. A component cluster of $\Lambda$ is the set of vertices of a maximal complete subgraph of the graph of strongly reduced irreducible components. It is $E$-rigid if $E_\Lambda(Z)=0$ for every component $Z$ in the cluster.

Component-cluster conjecture. For any basic algebra $\Lambda$:

$$\text{(i) every component cluster of }\Lambda\text{ has cardinality at most }n;$$ $$\text{(ii) the $E$-rigid component clusters are exactly the component clusters of cardinality }n.$$

The paper notes that this may be too optimistic, although it holds for the path algebra of an acyclic quiver. The conjecture concerns the size and rigidity of component clusters in the general theory of Caldero–Chapoton algebras.