Lee–Lee conjecture on non-self-crossing admissible curves and real Schur roots

Let $Q$ be an acyclic quiver. An admissible curve is a certain path on the Riemann surface associated to $Q$, and each such curve has an associated root. A non-self-crossing admissible curve is an admissible curve without self-intersections. The real Schur roots of $Q$ are the dimension vectors of indecomposable exceptional modules of the path algebra of $Q$.

Lee–Lee conjecture. The set of roots associated to non-self-crossing admissible curves and the set of real Schur roots coincide.

This conjecture gives a diagrammatic description of real Schur roots, and hence of positive $c$-vectors, for acyclic quivers. It is known for acyclic quivers of finite type, but the general acyclic case is presented here as a conjecture.