Realization by reflections for mutation classes from unpunctured surfaces or orbifolds

Let a mutation class be originating from an unpunctured surface or orbifold if it arises from an unpunctured surface or orbifold in the sense of the paper. A realization by reflections assigns vectors and reflections to the quiver data so that the associated Gram matrix and exchange matrix satisfy the required realization conditions. Realization conjecture. Every mutation class originating from an unpunctured surface or orbifold admits a realization by reflections. The preceding discussion contrasts this with mutation classes originating from punctured surfaces or orbifolds, which in the non-acyclic case do not possess such an admissible realization; the conjectured unpunctured case is presented as supported by strong evidence, but its general status is not established.