Bang-Jensen–Gutin–Li hamiltonicity conjecture for dominating pairs

Let $D$ be a finite strong digraph without loops or multiple arcs on $n$ vertices. For distinct vertices $x,y$, write $x\rightarrow y$ when there is an arc from $x$ to $y$. A pair of vertices is dominating if both vertices send an arc to a common vertex, and a pair is dominated if a common vertex sends an arc to both of them. Here $d(x)$ denotes the degree of $x$.

Bang-Jensen–Gutin–Li conjecture. If

$$d(x)+d(y)\geq 2n-1$$

for every pair of dominating non-adjacent vertices and every pair of dominated non-adjacent vertices $\{x,y\}$, then $D$ is hamiltonian.

This conjecture was raised by Bang-Jensen, Gutin and Li as a degree-sum condition combining local domination structure with hamiltonicity. It is still open and seems quite difficult.