Uniqueness conjecture for graphs with cyclic defect or excess

A graph with defect or excess $2$ is said to have cyclic defect or excess when its defect or excess matrix is the adjacency matrix of a cycle. The Möbius ladder on $8$ vertices is the exceptional graph identified in the paper. Cyclic defect or excess uniqueness conjecture. Apart from the Möbius ladder on $8$ vertices, there is no graph with cyclic defect or excess. The authors note that the structural constraints imposed by cyclic defect or excess lead them to believe that this Möbius ladder is the only example; the paper establishes substantial partial results, including finiteness in odd degree, but not the full conjecture.