Pancyclicity conjecture for Bang–Jensen–Gutin–Li type conditions

A digraph $D$ has order $n\ge 3$. A pair of nonadjacent vertices is dominated if it has a common in-neighbor, and dominating if it has a common out-neighbor. Let $d(u)$ denote the total degree of a vertex $u$, and let $\mathcal{D}_L$ be the class of non-pancyclic locally semicomplete digraphs defined by the stated round-decomposition characterization. Let $\mathcal{D}_B$ be the class of balanced complete bipartite digraphs with at least $4$ vertices. Pancyclicity conjecture. If every vertex $u\in V(D)$ belonging to a nonadjacent dominated pair or a nonadjacent dominating pair satisfies $d(u)\ge n$, then $D$ is pancyclic unless $D\in\mathcal{D}_L$ or $D\in\mathcal{D}_B$. This conjecture seeks a pancyclicity criterion extending Bang–Jensen–Gutin–Li type Hamiltonicity conditions; the exceptional classes account for known non-pancyclic structures, while a complete characterization of non-pancyclic locally semicomplete digraphs under the relevant condition remains difficult.