Total regularity conjecture for mixed graphs with defect or excess one

Let an $(r,z,k;-1)$-graph be a mixed graph with directed degree $r$, undirected degree $z$, diameter $k$, and defect one; let an $(r,z,k;+1)$-graph be a mixed graph with the same degree and diameter parameters and excess one. A mixed graph is totally regular when every vertex has the same indegree and outdegree, and every vertex has the same number of undirected neighbours.

Total regularity conjecture. All $(r,z,k;-1)$- and $(r,z,k;+1)$-graphs are totally regular.

The paper establishes total regularity for diameter two and defect one, and for $2$-geodetic mixed graphs with excess one. The analogous question for larger values of $k$ remains largely open.