The nebula conjecture

Let $T$ be a tournament. A nebula is a tournament admitting an ordering whose vertices are partitioned into the vertex sets of stars and singleton components, with no restriction on whether a star is left or right and no condition on the locations of star centers. Nebula conjecture. Every nebula satisfies the Erdős–Hajnal conjecture; equivalently, for every nebula $H$, there exists $\epsilon(H)>0$ such that every $H$-free tournament on $n$ vertices contains a transitive subtournament of size at least $n^{\epsilon(H)}$. Galaxies are known to satisfy the Erdős–Hajnal conjecture, but it is not known whether the result extends to nebulae after abandoning the galaxy condition on star centers.