Classification of orientations admitting Ringel self-duality

Determine the full list of orientations of Dynkin diagrams of type D_{2m} and A_{4m+1} for which there exists a partial ordering on the vertices such that the corresponding path algebra with its quasi-hereditary structure is Ringel self-dual.

Background

Theorem 4.1 establishes that the underlying Dynkin diagram of a hereditary path algebra admitting a Ringel-self-dual quasi-hereditary structure must be of type D_{2m} or A_{4m+1}, and the paper constructs examples for both families. However, the classification does not determine which orientations within these underlying graphs support such a structure. The unresolved problem is to give the complete orientation-level classification, together with the existence of a compatible partial order.